Theorem ifbieq12i 4510

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 4486	. . 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 4508	. 2 ⊢ if(𝜑, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷)
7	3, 6	eqtri 2783	1 ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  ifcif 4482

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-un 3904  df-if 4483

This theorem is used by:  sgnneg  15176  cbvditg  26084  nosupcbv  27941  noinfcbv  27956  ditgeq123i  36832  cbvditgvw2  36872  binomcxplemdvsum  45182