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Theorem cleq2lem 36933
 Description: Equality implies bijection. (Contributed by RP, 24-Jul-2020.)
Hypothesis
Ref Expression
cleq2lem.b (𝐴 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
cleq2lem (𝐴 = 𝐵 → ((𝑅𝐴𝜑) ↔ (𝑅𝐵𝜓)))

Proof of Theorem cleq2lem
StepHypRef Expression
1 sseq2 3590 . 2 (𝐴 = 𝐵 → (𝑅𝐴𝑅𝐵))
2 cleq2lem.b . 2 (𝐴 = 𝐵 → (𝜑𝜓))
31, 2anbi12d 743 1 (𝐴 = 𝐵 → ((𝑅𝐴𝜑) ↔ (𝑅𝐵𝜓)))
 Colors of variables: wff setvar class 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
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