Theorem ifeq12da 4514

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 4512	. . 3 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐵))
3		iftrue 4486	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
4		iftrue 4486	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐶)
5	3, 4	eqtr4d 2800	. . 3 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷))
6	2, 5	sylan9eq 2817	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
7		ifeq12da.2	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)
8	7	ifeq2da 4513	. . 3 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐷))
9		iffalse 4489	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐷) = 𝐷)
10		iffalse 4489	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐷)
11	9, 10	eqtr4d 2800	. . 3 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐷) = if(𝜓, 𝐶, 𝐷))
12	8, 11	sylan9eq 2817	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
13	6, 12	pm2.61dan 822	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1560  ifcif 4480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-un 3909  df-if 4481

This theorem is referenced by:  ifbieq12d2  4515  psdmvr  22234  copco  25080  pcohtpylem  25081  rpvmasum2  27576  mplmulmvr  33836  prjspnfv01  43206