Theorem ifeq2da 4560

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 4534	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = 𝐶)
2		iftrue 4534	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
3	1, 2	eqtr4d 2775	. . 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 4548	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
7	4, 6	pm2.61dan 811	1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541  ifcif 4528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-un 3953  df-if 4529

This theorem is referenced by:  ifeq12da  4561  dfac12lem1  10140  ttukeylem3  10508  xmulcom  13249  xmulneg1  13252  subgmulg  19056  1marepvmarrepid  22297  pcopt2  24763  limcdif  25617  limcmpt  25624  limcres  25627  limccnp  25632  radcnv0  26152  leibpi  26671  efrlim  26698  dchrvmasumiflem2  27229  padicabvf  27358  padicabvcxp  27359  itg2addnclem  36842  fourierdlem73  45194  fourierdlem76  45197  fourierdlem89  45210  fourierdlem91  45212