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Theorem ineq12d 3278
 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 3272 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
41, 2, 3syl2anc 408 1 (𝜑 → (𝐴𝐶) = (𝐵𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ∩ cin 3070 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 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-v 2688  df-in 3077 This theorem is referenced by:  csbing  3283  funprg  5173  funtpg  5174  offval  5989  ofrfval  5990  undifdc  6812  djudom  6978  ressid2  12027  ressval2  12028
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