Theorem ifnetruedc 3626

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 1002	. 2 ⊢ ((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) → DECID 𝜑)
2		iffalse 3590	. . . 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 1007	. . . . 5 ⊢ (((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → if(𝜑, 𝐴, 𝐵) = 𝐴)
5		simpl2 1006	. . . . 5 ⊢ (((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → 𝐴 ≠ 𝐵)
6	4, 5	eqnetrd 2404	. . . 4 ⊢ (((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → if(𝜑, 𝐴, 𝐵) ≠ 𝐵)
7	6	neneqd 2401	. . 3 ⊢ (((DECID 𝜑 ∧ 𝐴 ≠ 𝐵 ∧ if(𝜑, 𝐴, 𝐵) = 𝐴) ∧ ¬ 𝜑) → ¬ if(𝜑, 𝐴, 𝐵) = 𝐵)
8	3, 7	condandc 885	. 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 838   ∧ w3a 983   = wceq 1375   ≠ wne 2380  ifcif 3582

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-11 1532  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-dc 839  df-3an 985  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-ne 2381  df-if 3583

This theorem is referenced by:  ifnebibdc  3628