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Theorem ineq12d 3133
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 3127 . 2 ((A = B 𝐶 = 𝐷) → (A𝐶) = (B𝐷))
41, 2, 3syl2anc 391 1 (φ → (A𝐶) = (B𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  cin 2910
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918
This theorem is referenced by:  csbing  3138  funprg  4892  funtpg  4893  offval  5661  ofrfval  5662
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