Theorem cbvcllem 44801

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 44800	. 2 ⊢ (𝑥 = 𝑦 → ((𝑋 ⊆ 𝑥 ∧ 𝜑) ↔ (𝑋 ⊆ 𝑦 ∧ 𝜓)))
3	2	cbvabv 2834	1 ⊢ {𝑥 ∣ (𝑋 ⊆ 𝑥 ∧ 𝜑)} = {𝑦 ∣ (𝑋 ⊆ 𝑦 ∧ 𝜓)}

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  {cab 2741   ⊆ wss 3900

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 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-ss 3901

This theorem is used by: (None)