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Theorem seeq2 5421
Description: Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
seeq2 (𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))

Proof of Theorem seeq2
StepHypRef Expression
1 eqimss2 3949 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 sess2 5417 . . 3 (𝐵𝐴 → (𝑅 Se 𝐴𝑅 Se 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))
4 eqimss 3948 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 sess2 5417 . . 3 (𝐴𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
73, 6impbid 213 1 (𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1522  wss 3863   Se wse 5405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5099
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rab 3114  df-v 3439  df-in 3870  df-ss 3878  df-se 5408
This theorem is referenced by:  oieq2  8828
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