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Theorem cleq2lem 43598
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 4022 . 2 (𝐴 = 𝐵 → (𝑅𝐴𝑅𝐵))
2 cleq2lem.b . 2 (𝐴 = 𝐵 → (𝜑𝜓))
31, 2anbi12d 632 1 (𝐴 = 𝐵 → ((𝑅𝐴𝜑) ↔ (𝑅𝐵𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wss 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-ss 3980
This theorem is referenced by:  cbvcllem  43599  clublem  43600  rclexi  43605  rtrclex  43607  rtrclexi  43611  clrellem  43612  clcnvlem  43613  trcleq2lemRP  43620  dfrcl2  43664  brtrclfv2  43717  clsk1indlem1  44035
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