Theorem ifeq1dadc 3545

Description: Conditional equality. (Contributed by Jim Kingdon, 1-Jan-2022.)

Hypotheses

Ref	Expression
ifeq1dadc.1	⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵)
ifeq1dadc.dc	⊢ (𝜑 → DECID 𝜓)

Assertion

Ref	Expression
ifeq1dadc	⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Proof of Theorem ifeq1dadc

Step	Hyp	Ref	Expression
1		ifeq1dadc.1	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐵)
2	1	ifeq1d 3532	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
3		iffalse 3523	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = 𝐶)
4		iffalse 3523	. . . 4 ⊢ (¬ 𝜓 → if(𝜓, 𝐵, 𝐶) = 𝐶)
5	3, 4	eqtr4d 2200	. . 3 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
6	5	adantl 275	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
7		ifeq1dadc.dc	. . 3 ⊢ (𝜑 → DECID 𝜓)
8		exmiddc 826	. . 3 ⊢ (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓))
9	7, 8	syl 14	. 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓))
10	2, 6, 9	mpjaodan 788	1 ⊢ (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 698  DECID wdc 824   = wceq 1342  ifcif 3515

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-dc 825  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rab 2451  df-v 2723  df-un 3115  df-if 3516

This theorem is referenced by:  sumeq2  11286  isumss  11318  prodeq2  11484