Theorem cbviotav 6096

Description: Change bound variables in a description binder. (Contributed by Andrew Salmon, 1-Aug-2011.)

Hypothesis

Ref	Expression
cbviotav.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbviotav	⊢ (℩𝑥𝜑) = (℩𝑦𝜓)

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥

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

Proof of Theorem cbviotav

Step	Hyp	Ref	Expression
1		cbviotav.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2		nfv 2013	. 2 ⊢ Ⅎ𝑦𝜑
3		nfv 2013	. 2 ⊢ Ⅎ𝑥𝜓
4	1, 2, 3	cbviota 6095	1 ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1656  ℩cio 6088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-sn 4400  df-uni 4661  df-iota 6090

This theorem is referenced by:  oeeui  7954  ellimciota  40635  fourierdlem96  41207  fourierdlem97  41208  fourierdlem98  41209  fourierdlem99  41210  fourierdlem105  41216  fourierdlem106  41217  fourierdlem108  41219  fourierdlem110  41221  fourierdlem112  41223  fourierdlem113  41224  fourierdlem115  41226  funressndmafv2rn  42119