Theorem cbviotav 6530

Description: Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 2404. Use the weaker cbviotavw 6528 when possible. (Contributed by Andrew Salmon, 1-Aug-2011.) (New usage is discouraged.)

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 1949	. 2 ⊢ Ⅎ𝑦𝜑
3		nfv 1949	. 2 ⊢ Ⅎ𝑥𝜓
4	1, 2, 3	cbviota 6529	1 ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ℩cio 6518

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-13 2404  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-sn 4589  df-uni 4873  df-iota 6520

This theorem is used by: (None)