Theorem ifeq1da 3688

Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypothesis

Ref	Expression
ifeq1da.1	⊢ ((φ ∧ ψ) → A = B)

Assertion

Ref	Expression
ifeq1da	⊢ (φ → if(ψ, A, C) = if(ψ, B, C))

Proof of Theorem ifeq1da

Step	Hyp	Ref	Expression
1		ifeq1da.1	. . 3 ⊢ ((φ ∧ ψ) → A = B)
2	1	ifeq1d 3677	. 2 ⊢ ((φ ∧ ψ) → if(ψ, A, C) = if(ψ, B, C))
3		iffalse 3670	. . . 4 ⊢ (¬ ψ → if(ψ, A, C) = C)
4		iffalse 3670	. . . 4 ⊢ (¬ ψ → if(ψ, B, C) = C)
5	3, 4	eqtr4d 2388	. . 3 ⊢ (¬ ψ → if(ψ, A, C) = if(ψ, B, C))
6	5	adantl 452	. 2 ⊢ ((φ ∧ ¬ ψ) → if(ψ, A, C) = if(ψ, B, C))
7	2, 6	pm2.61dan 766	1 ⊢ (φ → if(ψ, A, C) = if(ψ, B, C))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   = 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-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-if 3664

This theorem is referenced by: (None)