Theorem ifbieq12i 3684

Description: Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)

Hypotheses

Ref	Expression
ifbieq12i.1	⊢ (φ ↔ ψ)
ifbieq12i.2	⊢ A = C
ifbieq12i.3	⊢ B = D

Assertion

Ref	Expression
ifbieq12i	⊢ if(φ, A, B) = if(ψ, C, D)

Proof of Theorem ifbieq12i

Step	Hyp	Ref	Expression
1		ifbieq12i.2	. . 3 ⊢ A = C
2		ifeq1 3667	. . 3 ⊢ (A = C → if(φ, A, B) = if(φ, C, B))
3	1, 2	ax-mp 5	. 2 ⊢ if(φ, A, B) = if(φ, C, B)
4		ifbieq12i.1	. . 3 ⊢ (φ ↔ ψ)
5		ifbieq12i.3	. . 3 ⊢ B = D
6	4, 5	ifbieq2i 3682	. 2 ⊢ if(φ, C, B) = if(ψ, C, D)
7	3, 6	eqtri 2373	1 ⊢ if(φ, A, B) = if(ψ, C, D)

Syntax hints:   ↔ 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: (None)