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Theorem bnj1262 32106
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 3506 1  |-  ( ph  ->  C  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    C_ wss 3428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-in 3435  df-ss 3442
This theorem is referenced by:  bnj229  32179  bnj1128  32283  bnj1145  32286
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