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Theorem ineq12d 3458
 Description: Equality deduction for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Hypotheses
Ref Expression
ineq1d.1 (φA = B)
ineq12d.2 (φC = D)
Assertion
Ref Expression
ineq12d (φ → (AC) = (BD))

Proof of Theorem ineq12d
StepHypRef Expression
1 ineq1d.1 . 2 (φA = B)
2 ineq12d.2 . 2 (φC = D)
3 ineq12 3452 . 2 ((A = B C = D) → (AC) = (BD))
41, 2, 3syl2anc 642 1 (φ → (AC) = (BD))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∩ cin 3208 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213 This theorem is referenced by:  csbing  3462  nnsucelrlem3  4426  funprg  5149  funprgOLD  5150
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