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Theorem vtoclgaf 3257
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 17-Feb-2006.) (Revised by Mario Carneiro, 10-Oct-2016.)
Hypotheses
Ref Expression
vtoclgaf.1 𝑥𝐴
vtoclgaf.2 𝑥𝜓
vtoclgaf.3 (𝑥 = 𝐴 → (𝜑𝜓))
vtoclgaf.4 (𝑥𝐵𝜑)
Assertion
Ref Expression
vtoclgaf (𝐴𝐵𝜓)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem vtoclgaf
StepHypRef Expression
1 vtoclgaf.1 . . 3 𝑥𝐴
21nfel1 2775 . . . 4 𝑥 𝐴𝐵
3 vtoclgaf.2 . . . 4 𝑥𝜓
42, 3nfim 1822 . . 3 𝑥(𝐴𝐵𝜓)
5 eleq1 2686 . . . 4 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
6 vtoclgaf.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6imbi12d 334 . . 3 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
8 vtoclgaf.4 . . 3 (𝑥𝐵𝜑)
91, 4, 7, 8vtoclgf 3250 . 2 (𝐴𝐵 → (𝐴𝐵𝜓))
109pm2.43i 52 1 (𝐴𝐵𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wnf 1705  wcel 1987  wnfc 2748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188
This theorem is referenced by:  vtoclga  3258  ssiun2s  4530  iunopeqop  4941  fvmptss  6249  fvmptf  6257  fmptco  6351  tfis  7001  inar1  9541  sumss  14388  fprodn0  14634  prmind2  15322  lss1d  18882  itg2splitlem  23421  dgrle  23903  cnlnadjlem5  28779  poimirlem25  33066  stoweidlem26  39550
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