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Theorem bnj1262 29196
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1262.1  |-  A  C_  B
bnj1262.2  |-  ( ph  ->  C  =  A )
Assertion
Ref Expression
bnj1262  |-  ( ph  ->  C  C_  B )

Proof of Theorem bnj1262
StepHypRef Expression
1 bnj1262.2 . 2  |-  ( ph  ->  C  =  A )
2 bnj1262.1 . 2  |-  A  C_  B
31, 2syl6eqss 3492 1  |-  ( ph  ->  C  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    C_ wss 3414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-in 3421  df-ss 3428
This theorem is referenced by:  bnj229  29269  bnj1128  29373  bnj1145  29376
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