Theorem ifeq12da 4526

Description: Equivalence deduction for conditional operators. (Contributed by Wolf Lammen, 24-Jun-2021.)

Hypotheses

Ref	Expression
ifeq12da.1	⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)
ifeq12da.2	⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)

Assertion

Ref	Expression
ifeq12da	⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))

Proof of Theorem ifeq12da

Step	Hyp	Ref	Expression
1		ifeq12da.1	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)
2	1	ifeq1da 4524	. . 3 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐵))
3		iftrue 4498	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
4		iftrue 4498	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐶)
5	3, 4	eqtr4d 2807	. . 3 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷))
6	2, 5	sylan9eq 2824	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
7		ifeq12da.2	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)
8	7	ifeq2da 4525	. . 3 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐷))
9		iffalse 4501	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐷) = 𝐷)
10		iffalse 4501	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐷)
11	9, 10	eqtr4d 2807	. . 3 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐷) = if(𝜓, 𝐶, 𝐷))
12	8, 11	sylan9eq 2824	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
13	6, 12	pm2.61dan 824	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = 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:  ifbieq12d2  4527  psdmvr  22301  copco  25146  pcohtpylem  25147  rpvmasum2  27642  mplmulmvr  33874  prjspnfv01  43282