Theorem cleq2lem 43621

Description: Equality implies bijection. (Contributed by RP, 24-Jul-2020.)

Hypothesis

Ref	Expression
cleq2lem.b	⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cleq2lem	⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ 𝜑) ↔ (𝑅 ⊆ 𝐵 ∧ 𝜓)))

Proof of Theorem cleq2lem

Step	Hyp	Ref	Expression
1		sseq2 4010	. 2 ⊢ (𝐴 = 𝐵 → (𝑅 ⊆ 𝐴 ↔ 𝑅 ⊆ 𝐵))
2		cleq2lem.b	. 2 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))
3	1, 2	anbi12d 632	1 ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ 𝜑) ↔ (𝑅 ⊆ 𝐵 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ⊆ wss 3951

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 2007  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-ss 3968

This theorem is referenced by:  cbvcllem  43622  clublem  43623  rclexi  43628  rtrclex  43630  rtrclexi  43634  clrellem  43635  clcnvlem  43636  trcleq2lemRP  43643  dfrcl2  43687  brtrclfv2  43740  clsk1indlem1  44058