Theorem cbvopab1v 5109

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,𝑧   𝜑,𝑧   𝜓,𝑥

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

Proof of Theorem cbvopab1v

Step	Hyp	Ref	Expression
1		nfv 1915	. 2 ⊢ Ⅎ𝑧𝜑
2		nfv 1915	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvopab1v.1	. 2 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvopab1 5105	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  {copab 5094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3863  df-sn 4523  df-pr 4525  df-op 4529  df-opab 5095

This theorem is referenced by: (None)