Theorem ifbieq2d 3683

Description: Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)

Hypotheses

Ref	Expression
ifbieq2d.1	⊢ (φ → (ψ ↔ χ))
ifbieq2d.2	⊢ (φ → A = B)

Assertion

Ref	Expression
ifbieq2d	⊢ (φ → if(ψ, C, A) = if(χ, C, B))

Proof of Theorem ifbieq2d

Step	Hyp	Ref	Expression
1		ifbieq2d.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2	1	ifbid 3681	. 2 ⊢ (φ → if(ψ, C, A) = if(χ, C, A))
3		ifbieq2d.2	. . 3 ⊢ (φ → A = B)
4	3	ifeq2d 3678	. 2 ⊢ (φ → if(χ, C, A) = if(χ, C, B))
5	2, 4	eqtrd 2385	1 ⊢ (φ → if(ψ, C, A) = if(χ, C, B))

Syntax hints:   → wi 4   ↔ wb 176   = 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:  tfineq  4489