Theorem cleq2lem 41589

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 3958	. 2 ⊢ (𝐴 = 𝐵 → (𝑅 ⊆ 𝐴 ↔ 𝑅 ⊆ 𝐵))
2		cleq2lem.b	. 2 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))
3	1, 2	anbi12d 631	1 ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ 𝜑) ↔ (𝑅 ⊆ 𝐵 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ⊆ wss 3898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-in 3905  df-ss 3915

This theorem is referenced by:  cbvcllem  41590  clublem  41591  rclexi  41596  rtrclex  41598  rtrclexi  41602  clrellem  41603  clcnvlem  41604  trcleq2lemRP  41611  dfrcl2  41655  brtrclfv2  41708  clsk1indlem1  42028