Theorem ifsbdc 3532

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 826	. 2 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2		iftrue 3525	. . . . 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 3525	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐷)
6	4, 5	eqtr4d 2201	. . 3 ⊢ (𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸))
7		iffalse 3528	. . . . 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 3528	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐸)
11	9, 10	eqtr4d 2201	. . 3 ⊢ (¬ 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸))
12	6, 11	jaoi 706	. 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 698  DECID wdc 824   = wceq 1343  ifcif 3520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-dc 825  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-if 3521

This theorem is referenced by:  fvifdc  5508  lgsneg  13565  lgsdilem  13568