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Theorem cleq2lem 43621
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 4010 . 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 1540  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2729  df-ss 3968
This theorem is referenced by:  cbvcllem  43622  clublem  43623  rclexi  43628  rtrclex  43630  rtrclexi  43634  clrellem  43635  clcnvlem  43636  trcleq2lemRP  43643  dfrcl2  43687  brtrclfv2  43740  clsk1indlem1  44058
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