Theorem ifnetruedc 3646

Description: Deduce truth from a conditional operator value. (Contributed by Thierry Arnoux, 20-Feb-2025.)

Assertion

Ref	Expression
ifnetruedc	⊢ ((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)

Proof of Theorem ifnetruedc

Step	Hyp	Ref	Expression
1		simp1 1021	. 2 ⊢ ((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → DECID 𝜑)
2		iffalse 3610	. . . 4 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
3	2	adantl 277	. . 3 ⊢ (((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐵)
4		simpl3 1026	. . . . 5 ⊢ (((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐴)
5		simpl2 1025	. . . . 5 ⊢ (((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → 𝐴 ≠ 𝐵)
6	4, 5	eqnetrd 2424	. . . 4 ⊢ (((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → if(𝜑, 𝐴, 𝐵) ≠ 𝐵)
7	6	neneqd 2421	. . 3 ⊢ (((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → ¬ if(𝜑, 𝐴, 𝐵) = 𝐵)
8	3, 7	condandc 886	. 2 ⊢ (DECID 𝜑 → ((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑))
9	1, 8	mpcom 36	1 ⊢ ((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 839   ∧ w3a 1002   = wceq 1395   ≠ wne 2400  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-in1 617  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-3an 1004  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-ne 2401  df-if 3603

This theorem is referenced by:  ifnebibdc  3648