Theorem ifeq3da 30313

Description: Given an expression 𝐶 containing if(𝜓, 𝐸, 𝐹), substitute (hypotheses .1 and .2) and evaluate (hypotheses .3 and .4) it for both cases at the same time. (Contributed by Thierry Arnoux, 13-Dec-2021.)

Hypotheses

Ref	Expression
ifeq3da.1	⊢ (if(𝜓, 𝐸, 𝐹) = 𝐸 → 𝐶 = 𝐺)
ifeq3da.2	⊢ (if(𝜓, 𝐸, 𝐹) = 𝐹 → 𝐶 = 𝐻)
ifeq3da.3	⊢ (𝜑 → 𝐺 = 𝐴)
ifeq3da.4	⊢ (𝜑 → 𝐻 = 𝐵)

Assertion

Ref	Expression
ifeq3da	⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)

Proof of Theorem ifeq3da

Step	Hyp	Ref	Expression
1		iftrue 4431	. . . . 5 ⊢ (𝜓 → if(𝜓, 𝐸, 𝐹) = 𝐸)
2		ifeq3da.1	. . . . 5 ⊢ (if(𝜓, 𝐸, 𝐹) = 𝐸 → 𝐶 = 𝐺)
3	1, 2	syl 17	. . . 4 ⊢ (𝜓 → 𝐶 = 𝐺)
4	3	adantl 485	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 = 𝐺)
5		ifeq3da.3	. . . 4 ⊢ (𝜑 → 𝐺 = 𝐴)
6	5	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐺 = 𝐴)
7	4, 6	eqtr2d 2834	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)
8		iffalse 4434	. . . . 5 ⊢ (¬ 𝜓 → if(𝜓, 𝐸, 𝐹) = 𝐹)
9		ifeq3da.2	. . . . 5 ⊢ (if(𝜓, 𝐸, 𝐹) = 𝐹 → 𝐶 = 𝐻)
10	8, 9	syl 17	. . . 4 ⊢ (¬ 𝜓 → 𝐶 = 𝐻)
11	10	adantl 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐶 = 𝐻)
12		ifeq3da.4	. . . 4 ⊢ (𝜑 → 𝐻 = 𝐵)
13	12	adantr 484	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐻 = 𝐵)
14	11, 13	eqtr2d 2834	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐶)
15	7, 14	ifeqda 4460	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538  ifcif 4425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-if 4426

This theorem is referenced by:  circlemeth  32021