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Theorem cleq2lem 36830
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 3494 . 2 (𝐴 = 𝐵 → (𝑅𝐴𝑅𝐵))
2 cleq2lem.b . 2 (𝐴 = 𝐵 → (𝜑𝜓))
31, 2anbi12d 742 1 (𝐴 = 𝐵 → ((𝑅𝐴𝜑) ↔ (𝑅𝐵𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wss 3444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-in 3451  df-ss 3458
This theorem is referenced by:  cbvcllem  36831  clublem  36833  rclexi  36838  rtrclex  36840  rtrclexi  36844  clrellem  36845  clcnvlem  36846  trcleq2lemRP  36853  dfrcl2  36882  brtrclfv2  36935  clsk1indlem1  37260
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