Theorem 19.35-1 1561

Description: Forward direction of Theorem 19.35 of [Margaris] p. 90. The converse holds for classical logic but not (for all propositions) in intuitionistic logic (Contributed by Mario Carneiro, 2-Feb-2015.)

Assertion

Ref	Expression
19.35-1	⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem 19.35-1

Step	Hyp	Ref	Expression
1		19.29 1557	. . 3 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥(𝜑 → 𝜓)) → ∃𝑥(𝜑 ∧ (𝜑 → 𝜓)))
2		pm3.35 340	. . . 4 ⊢ ((𝜑 ∧ (𝜑 → 𝜓)) → 𝜓)
3	2	eximi 1537	. . 3 ⊢ (∃𝑥(𝜑 ∧ (𝜑 → 𝜓)) → ∃𝑥𝜓)
4	1, 3	syl 14	. 2 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥(𝜑 → 𝜓)) → ∃𝑥𝜓)
5	4	expcom 115	1 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1288  ∃wex 1427

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-ial 1473

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  19.35i  1562  19.25  1563  19.36-1  1609  19.37-1  1610  spimt  1672  sbequi  1768