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Theorem vtoclbg 2813
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 29-Apr-1994.)
Hypotheses
Ref Expression
vtoclbg.1 (𝑥 = 𝐴 → (𝜑𝜒))
vtoclbg.2 (𝑥 = 𝐴 → (𝜓𝜃))
vtoclbg.3 (𝜑𝜓)
Assertion
Ref Expression
vtoclbg (𝐴𝑉 → (𝜒𝜃))
Distinct variable groups:   𝑥,𝐴   𝜒,𝑥   𝜃,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝑉(𝑥)

Proof of Theorem vtoclbg
StepHypRef Expression
1 vtoclbg.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜒))
2 vtoclbg.2 . . 3 (𝑥 = 𝐴 → (𝜓𝜃))
31, 2bibi12d 235 . 2 (𝑥 = 𝐴 → ((𝜑𝜓) ↔ (𝜒𝜃)))
4 vtoclbg.3 . 2 (𝜑𝜓)
53, 4vtoclg 2812 1 (𝐴𝑉 → (𝜒𝜃))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1364  wcel 2160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754
This theorem is referenced by:  pm13.183  2890  sbc8g  2985  sbcco  2999  sbc5  3001  sbcie2g  3011  eqsbc1  3017  sbcng  3018  sbcimg  3019  sbcan  3020  sbcang  3021  sbcor  3022  sbcorg  3023  sbcbig  3024  sbcal  3029  sbcalg  3030  sbcex2  3031  sbcexg  3032  sbcel1v  3040  sbcralg  3056  sbcreug  3058  sbcel12g  3087  sbceqg  3088  csbiebg  3114  elpwg  3598  snssgOLD  3743  preq12bg  3788  elintg  3867  elintrabg  3872  sbcbrg  4072  opelresg  4932  elixpsn  6762  ixpsnf1o  6763  domeng  6779
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