Theorem cbvopab1v 4094

Description: Rule used to change the first bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Eric Schmidt, 4-Apr-2007.)

Hypothesis

Ref	Expression
cbvopab1v.1	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvopab1v	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}

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

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

Proof of Theorem cbvopab1v

Step	Hyp	Ref	Expression
1		nfv 1539	. 2 ⊢ Ⅎ𝑧𝜑
2		nfv 1539	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvopab1v.1	. 2 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvopab1 4091	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364  {copab 4078

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-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-sn 3613  df-pr 3614  df-op 3616  df-opab 4080

This theorem is referenced by: (None)