Theorem cbvopabv 4722

Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 15-Oct-1996.)

Hypothesis

Ref	Expression
cbvopabv.1	⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvopabv	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤   𝜑,𝑧,𝑤   𝜓,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑧,𝑤)

Proof of Theorem cbvopabv

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 ⊢ Ⅎ𝑧𝜑
2		nfv 1843	. 2 ⊢ Ⅎ𝑤𝜑
3		nfv 1843	. 2 ⊢ Ⅎ𝑥𝜓
4		nfv 1843	. 2 ⊢ Ⅎ𝑦𝜓
5		cbvopabv.1	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvopab 4721	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  {copab 4712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713

This theorem is referenced by:  cantnf  8590  infxpen  8837  axdc2  9271  fpwwe2cbv  9452  fpwwecbv  9466  sylow1  18018  bcth  23126  vitali  23382  lgsquadlem3  25107  lgsquad  25108  islnopp  25631  ishpg  25651  hpgbr  25652  trgcopy  25696  trgcopyeu  25698  acopyeu  25725  tgasa1  25739  axcontlem1  25844  eulerpartlemgvv  30438  eulerpart  30444  cvmlift2lem13  31297  pellex  37399  aomclem8  37631  sprsymrelf  41745