Theorem ifeq2dadc 3601

Description: Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
ifeq2da.1	⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)
ifeq2dadc.dc	⊢ (𝜑 → DECID 𝜓)

Assertion

Ref	Expression
ifeq2dadc	⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Proof of Theorem ifeq2dadc

Step	Hyp	Ref	Expression
1		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓)
2	1	iftrued 3577	. . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐶, 𝐴) = 𝐶)
3	1	iftrued 3577	. . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐶, 𝐵) = 𝐶)
4	2, 3	eqtr4d 2240	. 2 ⊢ ((𝜑 ∧ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
5		ifeq2da.1	. . 3 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)
6	5	ifeq2d 3588	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
7		ifeq2dadc.dc	. . 3 ⊢ (𝜑 → DECID 𝜓)
8		exmiddc 837	. . 3 ⊢ (DECID 𝜓 → (𝜓 ∨ ¬ 𝜓))
9	7, 8	syl 14	. 2 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓))
10	4, 6, 9	mpjaodan 799	1 ⊢ (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   = wceq 1372  ifcif 3570

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 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rab 2492  df-v 2773  df-un 3169  df-if 3571

This theorem is referenced by:  subgmulg  13442