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Theorem ifeq12d 3678
 Description: Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
Hypotheses
Ref Expression
ifeq1d.1 (φA = B)
ifeq12d.2 (φC = D)
Assertion
Ref Expression
ifeq12d (φ → if(ψ, A, C) = if(ψ, B, D))

Proof of Theorem ifeq12d
StepHypRef Expression
1 ifeq1d.1 . . 3 (φA = B)
21ifeq1d 3676 . 2 (φ → if(ψ, A, C) = if(ψ, B, C))
3 ifeq12d.2 . . 3 (φC = D)
43ifeq2d 3677 . 2 (φ → if(ψ, B, C) = if(ψ, B, D))
52, 4eqtrd 2385 1 (φ → if(ψ, A, C) = if(ψ, B, D))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = 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:  ifbieq12d  3684
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