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Theorem ineqan12d 3140
 Description: Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.)
Hypotheses
Ref Expression
ineq1d.1 (𝜑𝐴 = 𝐵)
ineqan12d.2 (𝜓𝐶 = 𝐷)
Assertion
Ref Expression
ineqan12d ((𝜑𝜓) → (𝐴𝐶) = (𝐵𝐷))

Proof of Theorem ineqan12d
StepHypRef Expression
1 ineq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 ineqan12d.2 . 2 (𝜓𝐶 = 𝐷)
3 ineq12 3133 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
41, 2, 3syl2an 273 1 ((𝜑𝜓) → (𝐴𝐶) = (𝐵𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∩ cin 2916 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924 This theorem is referenced by:  fvun1  5239  fndmin  5274  offval  5719  ofrfval  5720  offval3  5761
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