Theorem ifsbdc 3615

Description: Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.)

Hypotheses

Ref	Expression
ifsbdc.1	⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝐶 = 𝐷)
ifsbdc.2	⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → 𝐶 = 𝐸)

Assertion

Ref	Expression
ifsbdc	⊢ (DECID 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸))

Proof of Theorem ifsbdc

Step	Hyp	Ref	Expression
1		exmiddc 841	. 2 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2		iftrue 3607	. . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
3		ifsbdc.1	. . . . 5 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐴 → 𝐶 = 𝐷)
4	2, 3	syl 14	. . . 4 ⊢ (𝜑 → 𝐶 = 𝐷)
5		iftrue 3607	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐷)
6	4, 5	eqtr4d 2265	. . 3 ⊢ (𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸))
7		iffalse 3610	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
8		ifsbdc.2	. . . . 5 ⊢ (if(𝜑, 𝐴, 𝐵) = 𝐵 → 𝐶 = 𝐸)
9	7, 8	syl 14	. . . 4 ⊢ (¬ 𝜑 → 𝐶 = 𝐸)
10		iffalse 3610	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐸)
11	9, 10	eqtr4d 2265	. . 3 ⊢ (¬ 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸))
12	6, 11	jaoi 721	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) → 𝐶 = if(𝜑, 𝐷, 𝐸))
13	1, 12	syl 14	1 ⊢ (DECID 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 713  DECID wdc 839   = wceq 1395  ifcif 3602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-dc 840  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-if 3603

This theorem is referenced by:  fvifdc  5648  lgsneg  15697  lgsdilem  15700