NFE Home New Foundations Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  NFE Home  >  Th. List  >  eqeqan12d GIF version

Theorem eqeqan12d 2368
Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 9-Aug-1994.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypotheses
Ref Expression
eqeqan12d.1 (φA = B)
eqeqan12d.2 (ψC = D)
Assertion
Ref Expression
eqeqan12d ((φ ψ) → (A = CB = D))

Proof of Theorem eqeqan12d
StepHypRef Expression
1 eqeqan12d.1 . 2 (φA = B)
2 eqeqan12d.2 . 2 (ψC = D)
3 eqeq12 2365 . 2 ((A = B C = D) → (A = CB = D))
41, 2, 3syl2an 463 1 ((φ ψ) → (A = CB = D))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   wa 358   = wceq 1642
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-11 1746  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346
This theorem is referenced by:  eqeqan12rd  2369  adj11  3889  pw1equn  4331  pw1eqadj  4332  eqfnfv2  5393  pw1fnf1o  5855  dflec2  6210  tc11  6228
  Copyright terms: Public domain W3C validator