Theorem ifsb 3672

Description: Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)

Hypotheses

Ref	Expression
ifsb.1	⊢ ( if(φ, A, B) = A → C = D)
ifsb.2	⊢ ( if(φ, A, B) = B → C = E)

Assertion

Ref	Expression
ifsb	⊢ C = if(φ, D, E)

Proof of Theorem ifsb

Step	Hyp	Ref	Expression
1		iftrue 3669	. . . 4 ⊢ (φ → if(φ, A, B) = A)
2		ifsb.1	. . . 4 ⊢ ( if(φ, A, B) = A → C = D)
3	1, 2	syl 15	. . 3 ⊢ (φ → C = D)
4		iftrue 3669	. . 3 ⊢ (φ → if(φ, D, E) = D)
5	3, 4	eqtr4d 2388	. 2 ⊢ (φ → C = if(φ, D, E))
6		iffalse 3670	. . . 4 ⊢ (¬ φ → if(φ, A, B) = B)
7		ifsb.2	. . . 4 ⊢ ( if(φ, A, B) = B → C = E)
8	6, 7	syl 15	. . 3 ⊢ (¬ φ → C = E)
9		iffalse 3670	. . 3 ⊢ (¬ φ → if(φ, D, E) = E)
10	8, 9	eqtr4d 2388	. 2 ⊢ (¬ φ → C = if(φ, D, E))
11	5, 10	pm2.61i 156	1 ⊢ C = if(φ, D, E)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ifcif 3663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-if 3664

This theorem is referenced by:  fvif  5341