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Theorem ifeq2da 4497
Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
ifeq2da.1 ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)
Assertion
Ref Expression
ifeq2da (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Proof of Theorem ifeq2da
StepHypRef Expression
1 iftrue 4472 . . . 4 (𝜓 → if(𝜓, 𝐶, 𝐴) = 𝐶)
2 iftrue 4472 . . . 4 (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
31, 2eqtr4d 2859 . . 3 (𝜓 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
43adantl 484 . 2 ((𝜑𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
5 ifeq2da.1 . . 3 ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)
65ifeq2d 4485 . 2 ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
74, 6pm2.61dan 811 1 (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1533  ifcif 4466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-un 3940  df-if 4467
This theorem is referenced by:  ifeq12da  4498  dfac12lem1  9563  ttukeylem3  9927  xmulcom  12653  xmulneg1  12656  subgmulg  18287  1marepvmarrepid  21178  pcopt2  23621  limcdif  24468  limcmpt  24475  limcres  24478  limccnp  24483  radcnv0  24998  leibpi  25514  efrlim  25541  dchrvmasumiflem2  26072  padicabvf  26201  padicabvcxp  26202  itg2addnclem  34937  fourierdlem73  42458  fourierdlem76  42461  fourierdlem89  42474  fourierdlem91  42476
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