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Theorem br2ndeqgOLD 32001
Description: Obsolete version of br2ndeqg 7357 as of 9-Feb-2022. (Contributed by Scott Fenton, 2-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
br2ndeqgOLD ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))

Proof of Theorem br2ndeqgOLD
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 4553 . . . . . 6 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21breq1d 4814 . . . . 5 (𝑥 = 𝐴 → (⟨𝑥, 𝑦⟩2nd 𝐶 ↔ ⟨𝐴, 𝑦⟩2nd 𝐶))
32bibi1d 332 . . . 4 (𝑥 = 𝐴 → ((⟨𝑥, 𝑦⟩2nd 𝐶𝐶 = 𝑦) ↔ (⟨𝐴, 𝑦⟩2nd 𝐶𝐶 = 𝑦)))
43imbi2d 329 . . 3 (𝑥 = 𝐴 → ((𝐶𝑋 → (⟨𝑥, 𝑦⟩2nd 𝐶𝐶 = 𝑦)) ↔ (𝐶𝑋 → (⟨𝐴, 𝑦⟩2nd 𝐶𝐶 = 𝑦))))
5 opeq2 4554 . . . . . 6 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
65breq1d 4814 . . . . 5 (𝑦 = 𝐵 → (⟨𝐴, 𝑦⟩2nd 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶))
7 eqeq2 2771 . . . . 5 (𝑦 = 𝐵 → (𝐶 = 𝑦𝐶 = 𝐵))
86, 7bibi12d 334 . . . 4 (𝑦 = 𝐵 → ((⟨𝐴, 𝑦⟩2nd 𝐶𝐶 = 𝑦) ↔ (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵)))
98imbi2d 329 . . 3 (𝑦 = 𝐵 → ((𝐶𝑋 → (⟨𝐴, 𝑦⟩2nd 𝐶𝐶 = 𝑦)) ↔ (𝐶𝑋 → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))))
10 breq2 4808 . . . 4 (𝑧 = 𝐶 → (⟨𝑥, 𝑦⟩2nd 𝑧 ↔ ⟨𝑥, 𝑦⟩2nd 𝐶))
11 eqeq1 2764 . . . 4 (𝑧 = 𝐶 → (𝑧 = 𝑦𝐶 = 𝑦))
12 vex 3343 . . . . 5 𝑥 ∈ V
13 vex 3343 . . . . 5 𝑦 ∈ V
1412, 13br2ndeq 31999 . . . 4 (⟨𝑥, 𝑦⟩2nd 𝑧𝑧 = 𝑦)
1510, 11, 14vtoclbg 3407 . . 3 (𝐶𝑋 → (⟨𝑥, 𝑦⟩2nd 𝐶𝐶 = 𝑦))
164, 9, 15vtocl2g 3410 . 2 ((𝐴𝑉𝐵𝑊) → (𝐶𝑋 → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵)))
17163impia 1110 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1072   = wceq 1632  wcel 2139  cop 4327   class class class wbr 4804  2nd c2nd 7333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-2nd 7335
This theorem is referenced by: (None)
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