Theorem cbvopabv 4156

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 1574	. 2 ⊢ Ⅎ𝑧𝜑
2		nfv 1574	. 2 ⊢ Ⅎ𝑤𝜑
3		nfv 1574	. 2 ⊢ Ⅎ𝑥𝜓
4		nfv 1574	. 2 ⊢ Ⅎ𝑦𝜓
5		cbvopabv.1	. 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvopab 4155	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑤⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  {copab 4144

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4146

This theorem is referenced by:  lgsquadlem3  15758  lgsquad  15759