Theorem ifeq12da 4499

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 4497	. . 3 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐵))
3		iftrue 4473	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
4		iftrue 4473	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐶)
5	3, 4	eqtr4d 2859	. . 3 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷))
6	2, 5	sylan9eq 2876	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
7		ifeq12da.2	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)
8	7	ifeq2da 4498	. . 3 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐷))
9		iffalse 4476	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐷) = 𝐷)
10		iffalse 4476	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐷)
11	9, 10	eqtr4d 2859	. . 3 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐷) = if(𝜓, 𝐶, 𝐷))
12	8, 11	sylan9eq 2876	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
13	6, 12	pm2.61dan 811	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537  ifcif 4467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-un 3941  df-if 4468

This theorem is referenced by:  ifbieq12d2  4500  copco  23622  pcohtpylem  23623  rpvmasum2  26088