Theorem cbviotav 6506

Description: Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 2406. Use the weaker cbviotavw 6504 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 1947	. 2 ⊢ Ⅎ𝑦𝜑
3		nfv 1947	. 2 ⊢ Ⅎ𝑥𝜓
4	1, 2, 3	cbviota 6505	1 ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ℩cio 6494

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2406  ax-ext 2737

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-ss 3923  df-sn 4592  df-uni 4875  df-iota 6496

This theorem is used by: (None)