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Theorem ifeq1da 3687
 Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
ifeq1da.1 ((φ ψ) → A = B)
Assertion
Ref Expression
ifeq1da (φ → if(ψ, A, C) = if(ψ, B, C))

Proof of Theorem ifeq1da
StepHypRef Expression
1 ifeq1da.1 . . 3 ((φ ψ) → A = B)
21ifeq1d 3676 . 2 ((φ ψ) → if(ψ, A, C) = if(ψ, B, C))
3 iffalse 3669 . . . 4 ψ → if(ψ, A, C) = C)
4 iffalse 3669 . . . 4 ψ → if(ψ, B, C) = C)
53, 4eqtr4d 2388 . . 3 ψ → if(ψ, A, C) = if(ψ, B, C))
65adantl 452 . 2 ((φ ¬ ψ) → if(ψ, A, C) = if(ψ, B, C))
72, 6pm2.61dan 766 1 (φ → if(ψ, A, C) = if(ψ, B, C))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = wceq 1642   ifcif 3662 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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-if 3663 This theorem is referenced by: (None)
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