Theorem ifp2 986

Description: Forward direction of dfifp2dc 987. This direction does not require decidability. (Contributed by Jim Kingdon, 25-Jan-2026.)

Assertion

Ref	Expression
ifp2	⊢ (if-(𝜑, 𝜓, 𝜒) → ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))

Proof of Theorem ifp2

Step	Hyp	Ref	Expression
1		df-ifp 984	. 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
2		pm3.4 333	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 → 𝜓))
3		pm2.24 624	. . . . 5 ⊢ (𝜑 → (¬ 𝜑 → 𝜒))
4	3	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (¬ 𝜑 → 𝜒))
5	2, 4	jca 306	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
6		ax-in2 618	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
7	6	adantr 276	. . . 4 ⊢ ((¬ 𝜑 ∧ 𝜒) → (𝜑 → 𝜓))
8		pm3.4 333	. . . 4 ⊢ ((¬ 𝜑 ∧ 𝜒) → (¬ 𝜑 → 𝜒))
9	7, 8	jca 306	. . 3 ⊢ ((¬ 𝜑 ∧ 𝜒) → ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
10	5, 9	jaoi 721	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) → ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))
11	1, 10	sylbi 121	1 ⊢ (if-(𝜑, 𝜓, 𝜒) → ((𝜑 → 𝜓) ∧ (¬ 𝜑 → 𝜒)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 713  if-wif 983

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 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-ifp 984

This theorem is referenced by:  dfifp2dc  987