Theorem cbvoprab12v 5970

Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.)

Hypothesis

Ref	Expression
cbvoprab12v.1	⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvoprab12v	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}

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

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

Proof of Theorem cbvoprab12v

Step	Hyp	Ref	Expression
1		nfv 1539	. 2 ⊢ Ⅎ𝑤𝜑
2		nfv 1539	. 2 ⊢ Ⅎ𝑣𝜑
3		nfv 1539	. 2 ⊢ Ⅎ𝑥𝜓
4		nfv 1539	. 2 ⊢ Ⅎ𝑦𝜓
5		cbvoprab12v.1	. 2 ⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓))
6	1, 2, 3, 4, 5	cbvoprab12 5969	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  {coprab 5896

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-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-opab 4080  df-oprab 5899

This theorem is referenced by: (None)