 Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  coeq12d Structured version   GIF version

Theorem coeq12d 4443
 Description: Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
Hypotheses
Ref Expression
coeq12d.1 (φA = B)
coeq12d.2 (φ𝐶 = 𝐷)
Assertion
Ref Expression
coeq12d (φ → (A𝐶) = (B𝐷))

Proof of Theorem coeq12d
StepHypRef Expression
1 coeq12d.1 . . 3 (φA = B)
21coeq1d 4440 . 2 (φ → (A𝐶) = (B𝐶))
3 coeq12d.2 . . 3 (φ𝐶 = 𝐷)
43coeq2d 4441 . 2 (φ → (B𝐶) = (B𝐷))
52, 4eqtrd 2069 1 (φ → (A𝐶) = (B𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∘ ccom 4292 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-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810  df-co 4297 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator