Theorem ifeq12da 4564

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 4562	. . 3 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐵))
3		iftrue 4537	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = 𝐶)
4		iftrue 4537	. . . 4 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐶)
5	3, 4	eqtr4d 2778	. . 3 ⊢ (𝜓 → if(𝜓, 𝐶, 𝐵) = if(𝜓, 𝐶, 𝐷))
6	2, 5	sylan9eq 2795	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
7		ifeq12da.2	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)
8	7	ifeq2da 4563	. . 3 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐷))
9		iffalse 4540	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐷) = 𝐷)
10		iffalse 4540	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐶, 𝐷) = 𝐷)
11	9, 10	eqtr4d 2778	. . 3 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐷) = if(𝜓, 𝐶, 𝐷))
12	8, 11	sylan9eq 2795	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))
13	6, 12	pm2.61dan 813	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537  ifcif 4531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-un 3968  df-if 4532

This theorem is referenced by:  ifbieq12d2  4565  copco  25065  pcohtpylem  25066  rpvmasum2  27571  prjspnfv01  42611