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Theorem seeq2 5240
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 3800 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 sess2 5236 . . 3 (𝐵𝐴 → (𝑅 Se 𝐴𝑅 Se 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))
4 eqimss 3799 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 sess2 5236 . . 3 (𝐴𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
64, 5syl 17 . 2 (𝐴 = 𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
73, 6impbid 202 1 (𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  wss 3716   Se wse 5224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rab 3060  df-v 3343  df-in 3723  df-ss 3730  df-se 5227
This theorem is referenced by:  oieq2  8586
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