Theorem r19.35-1 2656

Description: Restricted quantifier version of 19.35-1 1647. (Contributed by Jim Kingdon, 4-Jun-2018.)

Assertion

Ref	Expression
r19.35-1	⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Proof of Theorem r19.35-1

Step	Hyp	Ref	Expression
1		r19.29 2643	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ (𝜑 → 𝜓)))
2		pm3.35 347	. . . 4 ⊢ ((𝜑 ∧ (𝜑 → 𝜓)) → 𝜓)
3	2	reximi 2603	. . 3 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ (𝜑 → 𝜓)) → ∃𝑥 ∈ 𝐴 𝜓)
4	1, 3	syl 14	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜑 ∧ ∃𝑥 ∈ 𝐴 (𝜑 → 𝜓)) → ∃𝑥 ∈ 𝐴 𝜓)
5	4	expcom 116	1 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∀𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐴 𝜓))

Syntax hints:   → wi 4   ∧ wa 104  ∀wral 2484  ∃wrex 2485

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-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-ral 2489  df-rex 2490

This theorem is referenced by:  r19.36av  2657  r19.37  2658  iinexgm  4198  bndndx  9294