Theorem ifbieq2i 3682

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

Hypotheses

Ref	Expression
ifbieq2i.1	⊢ (φ ↔ ψ)
ifbieq2i.2	⊢ A = B

Assertion

Ref	Expression
ifbieq2i	⊢ if(φ, C, A) = if(ψ, C, B)

Proof of Theorem ifbieq2i

Step	Hyp	Ref	Expression
1		ifbieq2i.1	. . 3 ⊢ (φ ↔ ψ)
2		ifbi 3680	. . 3 ⊢ ((φ ↔ ψ) → if(φ, C, A) = if(ψ, C, A))
3	1, 2	ax-mp 5	. 2 ⊢ if(φ, C, A) = if(ψ, C, A)
4		ifbieq2i.2	. . 3 ⊢ A = B
5		ifeq2 3668	. . 3 ⊢ (A = B → if(ψ, C, A) = if(ψ, C, B))
6	4, 5	ax-mp 5	. 2 ⊢ if(ψ, C, A) = if(ψ, C, B)
7	3, 6	eqtri 2373	1 ⊢ if(φ, C, A) = if(ψ, C, B)

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:  ifbieq12i  3684