Theorem cbviotavw 6322

Description: Change bound variables in a description binder. Version of cbviotav 6324 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by Andrew Salmon, 1-Aug-2011.) (Revised by Gino Giotto, 26-Jan-2024.)

Hypothesis

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

Assertion

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

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

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

Proof of Theorem cbviotavw

Step	Hyp	Ref	Expression
1		cbviotavw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2		nfv 1915	. 2 ⊢ Ⅎ𝑦𝜑
3		nfv 1915	. 2 ⊢ Ⅎ𝑥𝜓
4	1, 2, 3	cbviotaw 6321	1 ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1537  ℩cio 6312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-in 3943  df-ss 3952  df-sn 4568  df-uni 4839  df-iota 6314

This theorem is referenced by:  oeeui  8228  ellimciota  41944  fourierdlem96  42536  fourierdlem97  42537  fourierdlem98  42538  fourierdlem99  42539  fourierdlem105  42545  fourierdlem106  42546  fourierdlem108  42548  fourierdlem110  42550  fourierdlem112  42552  fourierdlem113  42553  fourierdlem115  42555  funressndmafv2rn  43471