Theorem ifeq2da 4487

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

Step	Hyp	Ref	Expression
1		iftrue 4460	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = 𝐶)
2		iftrue 4460	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
3	1, 2	eqtr4d 2777	. . 3 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
4	3	adantl 482	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
5		ifeq2da.1	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)
6	5	ifeq2d 4475	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
7	4, 6	pm2.61dan 818	1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547  ifcif 4454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-un 3888  df-if 4455

This theorem is referenced by:  ifeq12da  4488  dfac12lem1  10057  ttukeylem3  10424  xmulcom  13209  xmulneg1  13212  subgmulg  19107  1marepvmarrepid  22558  pcopt2  25008  limcdif  25861  limcmpt  25868  limcres  25871  limccnp  25876  radcnv0  26399  leibpi  26924  efrlim  26951  dchrvmasumiflem2  27483  padicabvf  27612  padicabvcxp  27613  itg2addnclem  38038  fourierdlem73  46622  fourierdlem76  46625  fourierdlem89  46638  fourierdlem91  46640