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Theorem sbcrextOLD 3545
Description: Obsolete proof of sbcrext 3544 as of 7-Jul-2021. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) (Revised by NM, 18-Aug-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcrextOLD (𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
Distinct variable groups:   𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑦)

Proof of Theorem sbcrextOLD
StepHypRef Expression
1 sbcng 3509 . . . . 5 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑))
21adantr 480 . . . 4 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑))
3 sbcralt 3543 . . . . . 6 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑))
4 nfnfc1 2796 . . . . . . . . 9 𝑦𝑦𝐴
5 id 22 . . . . . . . . . 10 (𝑦𝐴𝑦𝐴)
6 nfcvd 2794 . . . . . . . . . 10 (𝑦𝐴𝑦V)
75, 6nfeld 2802 . . . . . . . . 9 (𝑦𝐴 → Ⅎ𝑦 𝐴 ∈ V)
84, 7nfan1 2106 . . . . . . . 8 𝑦(𝑦𝐴𝐴 ∈ V)
9 sbcng 3509 . . . . . . . . 9 (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
109adantl 481 . . . . . . . 8 ((𝑦𝐴𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
118, 10ralbid 3012 . . . . . . 7 ((𝑦𝐴𝐴 ∈ V) → (∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
1211ancoms 468 . . . . . 6 ((𝐴 ∈ V ∧ 𝑦𝐴) → (∀𝑦𝐵 [𝐴 / 𝑥] ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
133, 12bitrd 268 . . . . 5 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
1413notbid 307 . . . 4 ((𝐴 ∈ V ∧ 𝑦𝐴) → (¬ [𝐴 / 𝑥]𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
152, 14bitrd 268 . . 3 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑))
16 dfrex2 3025 . . . 4 (∃𝑦𝐵 𝜑 ↔ ¬ ∀𝑦𝐵 ¬ 𝜑)
1716sbcbii 3524 . . 3 ([𝐴 / 𝑥]𝑦𝐵 𝜑[𝐴 / 𝑥] ¬ ∀𝑦𝐵 ¬ 𝜑)
18 dfrex2 3025 . . 3 (∃𝑦𝐵 [𝐴 / 𝑥]𝜑 ↔ ¬ ∀𝑦𝐵 ¬ [𝐴 / 𝑥]𝜑)
1915, 17, 183bitr4g 303 . 2 ((𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
20 sbcex 3478 . . . . 5 ([𝐴 / 𝑥]𝑦𝐵 𝜑𝐴 ∈ V)
2120con3i 150 . . . 4 𝐴 ∈ V → ¬ [𝐴 / 𝑥]𝑦𝐵 𝜑)
2221adantr 480 . . 3 ((¬ 𝐴 ∈ V ∧ 𝑦𝐴) → ¬ [𝐴 / 𝑥]𝑦𝐵 𝜑)
23 sbcex 3478 . . . . . . 7 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
24232a1i 12 . . . . . 6 (𝑦𝐴 → (𝑦𝐵 → ([𝐴 / 𝑥]𝜑𝐴 ∈ V)))
254, 7, 24rexlimd2 3054 . . . . 5 (𝑦𝐴 → (∃𝑦𝐵 [𝐴 / 𝑥]𝜑𝐴 ∈ V))
2625con3rr3 151 . . . 4 𝐴 ∈ V → (𝑦𝐴 → ¬ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2726imp 444 . . 3 ((¬ 𝐴 ∈ V ∧ 𝑦𝐴) → ¬ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑)
2822, 272falsed 365 . 2 ((¬ 𝐴 ∈ V ∧ 𝑦𝐴) → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
2919, 28pm2.61ian 848 1 (𝑦𝐴 → ([𝐴 / 𝑥]𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 [𝐴 / 𝑥]𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wcel 2030  wnfc 2780  wral 2941  wrex 2942  Vcvv 3231  [wsbc 3468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-sbc 3469
This theorem is referenced by: (None)
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