Theorem cleq2lem 36933

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 3590	. 2 ⊢ (𝐴 = 𝐵 → (𝑅 ⊆ 𝐴 ↔ 𝑅 ⊆ 𝐵))
2		cleq2lem.b	. 2 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓))
3	1, 2	anbi12d 743	1 ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ 𝜑) ↔ (𝑅 ⊆ 𝐵 ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ⊆ wss 3540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-in 3547  df-ss 3554

This theorem is referenced by:  cbvcllem  36934  clublem  36936  rclexi  36941  rtrclex  36943  rtrclexi  36947  clrellem  36948  clcnvlem  36949  trcleq2lemRP  36956  dfrcl2  36985  brtrclfv2  37038  clsk1indlem1  37363