Theorem ifbieq12i 4524

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 4500	. . 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 4522	. 2 ⊢ if(𝜑, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷)
7	3, 6	eqtri 2753	1 ⊢ if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)

Syntax hints:   ↔ wb 206   = wceq 1540  ifcif 4496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3412  df-v 3457  df-un 3927  df-if 4497

This theorem is referenced by:  cbvditg  25762  nosupcbv  27621  noinfcbv  27636  sgnneg  32766  ditgeq123i  36194  cbvditgvw2  36234  binomcxplemdvsum  44316