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Theorem seeq2 4077
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 2998 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 sess2 4075 . . 3 (𝐵𝐴 → (𝑅 Se 𝐴𝑅 Se 𝐵))
31, 2syl 14 . 2 (𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))
4 eqimss 2997 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 sess2 4075 . . 3 (𝐴𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
64, 5syl 14 . 2 (𝐴 = 𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
73, 6impbid 120 1 (𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1243  wss 2917   Se wse 4066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-se 4070
This theorem is referenced by: (None)
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