Theorem cbvcllem 41106

Description: Change of bound variable in class of supersets of a with a property. (Contributed by RP, 24-Jul-2020.)

Hypothesis

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

Assertion

Ref	Expression
cbvcllem	⊢ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜑)} = {𝑦 ∣ (𝑋 ⊆ 𝑦 ∧ 𝜓)}

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

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

Proof of Theorem cbvcllem

Step	Hyp	Ref	Expression
1		cbvcllem.y	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	cleq2lem 41105	. 2 ⊢ (𝑥 = 𝑦 → ((𝑋 ⊆ 𝑥 ∧ 𝜑) ↔ (𝑋 ⊆ 𝑦 ∧ 𝜓)))
3	2	cbvabv 2812	1 ⊢ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜑)} = {𝑦 ∣ (𝑋 ⊆ 𝑦 ∧ 𝜓)}

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1539  {cab 2715   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900

This theorem is referenced by: (None)