Theorem ifeq2da 4538

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 4511	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = 𝐶)
2		iftrue 4511	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
3	1, 2	eqtr4d 2774	. . 3 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
4	3	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
5		ifeq2da.1	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)
6	5	ifeq2d 4526	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
7	4, 6	pm2.61dan 812	1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540  ifcif 4505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-un 3936  df-if 4506

This theorem is referenced by:  ifeq12da  4539  dfac12lem1  10163  ttukeylem3  10530  xmulcom  13287  xmulneg1  13290  subgmulg  19128  1marepvmarrepid  22518  pcopt2  24979  limcdif  25834  limcmpt  25841  limcres  25844  limccnp  25849  radcnv0  26382  leibpi  26909  efrlim  26936  efrlimOLD  26937  dchrvmasumiflem2  27470  padicabvf  27599  padicabvcxp  27600  itg2addnclem  37700  fourierdlem73  46175  fourierdlem76  46178  fourierdlem89  46191  fourierdlem91  46193