Theorem ifbieq12i 4520

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 4496	. . 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 4518	. 2 ⊢ if(𝜑, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷)
7	3, 6	eqtri 2792	1 ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)

Syntax hints:   ↔ wb 209   = wceq 1567  ifcif 4492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-un 3918  df-if 4493

This theorem is referenced by:  sgnneg  15137  cbvditg  25982  nosupcbv  27832  noinfcbv  27847  ditgeq123i  36610  cbvditgvw2  36650  binomcxplemdvsum  44957