Theorem cbvcllem 42815

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 42814	. 2 ⊢ (𝑥 = 𝑦 → ((𝑋 ⊆ 𝑥 ∧ 𝜑) ↔ (𝑋 ⊆ 𝑦 ∧ 𝜓)))
3	2	cbvabv 2797	1 ⊢ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜑)} = {𝑦 ∣ (𝑋 ⊆ 𝑦 ∧ 𝜓)}

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533  {cab 2701   ⊆ wss 3940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3947  df-ss 3957

This theorem is referenced by: (None)