Theorem ifeq2da 4337

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 4312	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = 𝐶)
2		iftrue 4312	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
3	1, 2	eqtr4d 2864	. . 3 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
4	3	adantl 475	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
5		ifeq2da.1	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)
6	5	ifeq2d 4325	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
7	4, 6	pm2.61dan 847	1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1656  ifcif 4306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-un 3803  df-if 4307

This theorem is referenced by:  ifeq12da  4338  dfac12lem1  9280  ttukeylem3  9648  xmulcom  12384  xmulneg1  12387  subgmulg  17959  1marepvmarrepid  20749  copco  23187  pcopt2  23192  limcdif  24039  limcmpt  24046  limcres  24049  limccnp  24054  radcnv0  24569  leibpi  25082  efrlim  25109  dchrvmasumiflem2  25604  rpvmasum2  25614  padicabvf  25733  padicabvcxp  25734  itg2addnclem  33997  fourierdlem73  41183  fourierdlem76  41186  fourierdlem89  41199  fourierdlem91  41201