Theorem ifsbdc 3547

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 836	. 2 ⊢ (DECID 𝜑 → (𝜑 ∨ ¬ 𝜑))
2		iftrue 3540	. . . . 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 3540	. . . 4 ⊢ (𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐷)
6	4, 5	eqtr4d 2213	. . 3 ⊢ (𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸))
7		iffalse 3543	. . . . 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 3543	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐷, 𝐸) = 𝐸)
11	9, 10	eqtr4d 2213	. . 3 ⊢ (¬ 𝜑 → 𝐶 = if(𝜑, 𝐷, 𝐸))
12	6, 11	jaoi 716	. 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 708  DECID wdc 834   = wceq 1353  ifcif 3535

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 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-dc 835  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-if 3536

This theorem is referenced by:  fvifdc  5538  lgsneg  14428  lgsdilem  14431