Theorem cbvriotav 7361

Description: Change bound variable in a restricted description binder. Usage of this theorem is discouraged because it depends on ax-13 2371. Use the weaker cbvriotavw 7357 when possible. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
cbvriotav	⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)

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

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

Proof of Theorem cbvriotav

Step	Hyp	Ref	Expression
1		nfv 1914	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1914	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvriotav.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvriota 7360	1 ⊢ (℩𝑥 ∈ 𝐴 𝜑) = (℩𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ℩crio 7346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-13 2371  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-ss 3934  df-sn 4593  df-uni 4875  df-iota 6467  df-riota 7347

This theorem is referenced by: (None)