Theorem ifeq2da 4522

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 4495	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = 𝐶)
2		iftrue 4495	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
3	1, 2	eqtr4d 2803	. . 3 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
4	3	adantl 487	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
5		ifeq2da.1	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)
6	5	ifeq2d 4510	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
7	4, 6	pm2.61dan 825	1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570  ifcif 4489

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-un 3911  df-if 4490

This theorem is used by:  ifeq12da  4523  dfac12lem1  10143  ttukeylem3  10510  xmulcom  13308  xmulneg1  13311  subgmulg  19251  1marepvmarrepid  22782  pcopt2  25233  limcdif  26086  limcmpt  26093  limcres  26096  limccnp  26101  radcnv0  26630  leibpi  27158  efrlim  27185  dchrvmasumiflem2  27717  padicabvf  27846  padicabvcxp  27847  itg2addnclem  38379  fourierdlem73  46951  fourierdlem76  46954  fourierdlem89  46967  fourierdlem91  46969