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Theorem seeq2 4262
 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 3152 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 sess2 4260 . . 3 (𝐵𝐴 → (𝑅 Se 𝐴𝑅 Se 𝐵))
31, 2syl 14 . 2 (𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))
4 eqimss 3151 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 sess2 4260 . . 3 (𝐴𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
64, 5syl 14 . 2 (𝐴 = 𝐵 → (𝑅 Se 𝐵𝑅 Se 𝐴))
73, 6impbid 128 1 (𝐴 = 𝐵 → (𝑅 Se 𝐴𝑅 Se 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ⊆ wss 3071   Se wse 4251 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-in 3077  df-ss 3084  df-se 4255 This theorem is referenced by: (None)
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