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Theorem csbunigVD 39650
Description: Virtual deduction proof of csbuni 4600. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. csbuni 4600 is csbunigVD 39650 without virtual deductions and was automatically derived from csbunigVD 39650.
1:: (   𝐴𝑉   ▶   𝐴𝑉   )
2:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑧𝑦𝑧 𝑦)   )
3:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦𝐵𝑦 𝐴 / 𝑥𝐵)   )
4:2,3: (   𝐴𝑉   ▶   (([𝐴 / 𝑥]𝑧𝑦 [𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
5:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧𝑦 𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵))   )
6:4,5: (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧𝑦 𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
7:6: (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥](𝑧 𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
8:7: (   𝐴𝑉   ▶   (∃𝑦[𝐴 / 𝑥](𝑧 𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
9:1: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧 𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵))   )
10:8,9: (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧 𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
11:10: (   𝐴𝑉   ▶   𝑧([𝐴 / 𝑥]𝑦( 𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
12:11: (   𝐴𝑉   ▶   {𝑧[𝐴 / 𝑥]𝑦( 𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦 𝑦𝐴 / 𝑥𝐵)}   )
13:1: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧 𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)}    )
14:12,13: (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧 𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦 𝑦𝐴 / 𝑥𝐵)}   )
15:: 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
16:15: 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦 𝐵)}
17:1,16: (   𝐴𝑉   ▶   [𝐴 / 𝑥] 𝐵 = {𝑧 𝑦(𝑧𝑦𝑦𝐵)}   )
18:1,17: (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}   )
19:14,18: (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = {𝑧 𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}   )
20:: 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦 𝑦𝐴 / 𝑥𝐵)}
21:19,20: (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵   )
qed:21: (𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
(Contributed by Alan Sare, 10-Nov-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
csbunigVD (𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)

Proof of Theorem csbunigVD
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idn1 39309 . . . . . . . . . . . . 13 (   𝐴𝑉   ▶   𝐴𝑉   )
2 sbcg 3651 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝑦𝑧𝑦))
31, 2e1a 39371 . . . . . . . . . . . 12 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑧𝑦𝑧𝑦)   )
4 sbcel2gOLD 39274 . . . . . . . . . . . . 13 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵))
51, 4e1a 39371 . . . . . . . . . . . 12 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵)   )
6 pm4.38 611 . . . . . . . . . . . . 13 ((([𝐴 / 𝑥]𝑧𝑦𝑧𝑦) ∧ ([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵)) → (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
76ex 397 . . . . . . . . . . . 12 (([𝐴 / 𝑥]𝑧𝑦𝑧𝑦) → (([𝐴 / 𝑥]𝑦𝐵𝑦𝐴 / 𝑥𝐵) → (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))))
83, 5, 7e11 39432 . . . . . . . . . . 11 (   𝐴𝑉   ▶   (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
9 sbcangOLD 39258 . . . . . . . . . . . 12 (𝐴𝑉 → ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵)))
101, 9e1a 39371 . . . . . . . . . . 11 (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵))   )
11 bibi1 340 . . . . . . . . . . . 12 (([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵)) → (([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)) ↔ (([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))))
1211biimprcd 240 . . . . . . . . . . 11 ((([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)) → (([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ([𝐴 / 𝑥]𝑧𝑦[𝐴 / 𝑥]𝑦𝐵)) → ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))))
138, 10, 12e11 39432 . . . . . . . . . 10 (   𝐴𝑉   ▶   ([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
1413gen11 39360 . . . . . . . . 9 (   𝐴𝑉   ▶   𝑦([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
15 exbi 1922 . . . . . . . . 9 (∀𝑦([𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ (𝑧𝑦𝑦𝐴 / 𝑥𝐵)) → (∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)))
1614, 15e1a 39371 . . . . . . . 8 (   𝐴𝑉   ▶   (∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
17 sbcexgOLD 39272 . . . . . . . . 9 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵)))
181, 17e1a 39371 . . . . . . . 8 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵))   )
19 bibi1 340 . . . . . . . . 9 (([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵)) → (([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)) ↔ (∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))))
2019biimprcd 240 . . . . . . . 8 ((∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)) → (([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦[𝐴 / 𝑥](𝑧𝑦𝑦𝐵)) → ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))))
2116, 18, 20e11 39432 . . . . . . 7 (   𝐴𝑉   ▶   ([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
2221gen11 39360 . . . . . 6 (   𝐴𝑉   ▶   𝑧([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵))   )
23 abbi 2885 . . . . . . 7 (∀𝑧([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)) ↔ {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
2423biimpi 206 . . . . . 6 (∀𝑧([𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵) ↔ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)) → {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)})
2522, 24e1a 39371 . . . . 5 (   𝐴𝑉   ▶   {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}   )
26 csbabgOLD 39569 . . . . . 6 (𝐴𝑉𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)})
271, 26e1a 39371 . . . . 5 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)}   )
28 eqeq2 2781 . . . . . 6 ({𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} → (𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} ↔ 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}))
2928biimpd 219 . . . . 5 ({𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} → (𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧[𝐴 / 𝑥]𝑦(𝑧𝑦𝑦𝐵)} → 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}))
3025, 27, 29e11 39432 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}   )
31 df-uni 4573 . . . . . . 7 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
3231ax-gen 1869 . . . . . 6 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}
33 spsbc 3598 . . . . . 6 (𝐴𝑉 → (∀𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} → [𝐴 / 𝑥] 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}))
341, 32, 33e10 39438 . . . . 5 (   𝐴𝑉   ▶   [𝐴 / 𝑥] 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}   )
35 sbceqg 4126 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥] 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} ↔ 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}))
3635biimpd 219 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥] 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} → 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}))
371, 34, 36e11 39432 . . . 4 (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)}   )
38 eqeq2 2781 . . . . 5 (𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} → (𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} ↔ 𝐴 / 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}))
3938biimpd 219 . . . 4 (𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} → (𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥{𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐵)} → 𝐴 / 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}))
4030, 37, 39e11 39432 . . 3 (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}   )
41 df-uni 4573 . . 3 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}
42 eqeq2 2781 . . . 4 ( 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} → (𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵𝐴 / 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)}))
4342biimprcd 240 . . 3 (𝐴 / 𝑥 𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} → ( 𝐴 / 𝑥𝐵 = {𝑧 ∣ ∃𝑦(𝑧𝑦𝑦𝐴 / 𝑥𝐵)} → 𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵))
4440, 41, 43e10 39438 . 2 (   𝐴𝑉   ▶   𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵   )
4544in1 39306 1 (𝐴𝑉𝐴 / 𝑥 𝐵 = 𝐴 / 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1628   = wceq 1630  wex 1851  wcel 2144  {cab 2756  [wsbc 3585  csb 3680   cuni 4572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-nul 4062  df-uni 4573  df-vd1 39305
This theorem is referenced by: (None)
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