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Theorem vtoclgaf 3535
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 2920 . . . 4 𝑥 𝐴𝐵
3 vtoclgaf.2 . . . 4 𝑥𝜓
42, 3nfim 1900 . . 3 𝑥(𝐴𝐵𝜓)
5 eleq1 2822 . . . 4 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
6 vtoclgaf.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
75, 6imbi12d 345 . . 3 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
8 vtoclgaf.4 . . 3 (𝑥𝐵𝜑)
91, 4, 7, 8vtoclgf 3525 . 2 (𝐴𝐵 → (𝐴𝐵𝜓))
109pm2.43i 52 1 (𝐴𝐵𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wnf 1786  wcel 2107  wnfc 2884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-v 3449
This theorem is referenced by:  ssiun2s  5012  iunopeqop  5482  fvmptss  6964  fvmptf  6973  fmptco  7079  tfis  7795  inar1  10719  sumss  15617  fprodn0  15870  prmind2  16569  lss1d  20468  itg2splitlem  25136  dgrle  25627  cnlnadjlem5  31062  poimirlem25  36153  stoweidlem26  44357
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