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Theorem vtocl2g 3573
Description: Implicit substitution of 2 classes for 2 setvar variables. (Contributed by NM, 25-Apr-1995.) Remove dependency on ax-10 2140, ax-11 2156, and ax-13 2376. (Revised by Steven Nguyen, 29-Nov-2022.)
Hypotheses
Ref Expression
vtocl2g.1 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl2g.2 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl2g.3 𝜑
Assertion
Ref Expression
vtocl2g ((𝐴𝑉𝐵𝑊) → 𝜒)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴   𝑦,𝐵   𝜓,𝑥   𝜒,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝐵(𝑥)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem vtocl2g
StepHypRef Expression
1 elex 3500 . 2 (𝐴𝑉𝐴 ∈ V)
2 vtocl2g.2 . . . 4 (𝑦 = 𝐵 → (𝜓𝜒))
32imbi2d 340 . . 3 (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒)))
4 vtocl2g.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
5 vtocl2g.3 . . . 4 𝜑
64, 5vtoclg 3553 . . 3 (𝐴 ∈ V → 𝜓)
73, 6vtoclg 3553 . 2 (𝐵𝑊 → (𝐴 ∈ V → 𝜒))
81, 7mpan9 506 1 ((𝐴𝑉𝐵𝑊) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  Vcvv 3479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481
This theorem is referenced by:  vtocl3g  3574  vtocl4g  3584  opthg  5481  opelopabsb  5534  vtoclr  5747  funopg  6599  f1osng  6888  fsng  7156  fnpr2g  7231  unexbOLD  7769  op1stg  8027  op2ndg  8028  xpsneng  9097  xpcomeng  9105  sbth  9134  sbthfi  9240  unxpdom  9290  prcdnq  11034  mhmlem  19081  carsgmon  34317  brimageg  35929  brdomaing  35937  brrangeg  35938  rankung  36168  mbfresfi  37674  zindbi  42963  2sbc6g  44439  2sbc5g  44440  fmulcl  45601
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