Theorem cbviotav 6466

Description: Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker cbviotavw 6464 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 1916	. 2 ⊢ Ⅎ𝑦𝜑
3		nfv 1916	. 2 ⊢ Ⅎ𝑥𝜓
4	1, 2, 3	cbviota 6465	1 ⊢ (℩𝑥𝜑) = (℩𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542  ℩cio 6454

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-13 2377  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-ss 3920  df-sn 4583  df-uni 4866  df-iota 6456

This theorem is referenced by: (None)