Theorem 19.35-1 1647

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 1643	. . 3 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥(𝜑 → 𝜓)) → ∃𝑥(𝜑 ∧ (𝜑 → 𝜓)))
2		pm3.35 347	. . . 4 ⊢ ((𝜑 ∧ (𝜑 → 𝜓)) → 𝜓)
3	2	eximi 1623	. . 3 ⊢ (∃𝑥(𝜑 ∧ (𝜑 → 𝜓)) → ∃𝑥𝜓)
4	1, 3	syl 14	. 2 ⊢ ((∀𝑥𝜑 ∧ ∃𝑥(𝜑 → 𝜓)) → ∃𝑥𝜓)
5	4	expcom 116	1 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1371  ∃wex 1515

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

This theorem is referenced by:  19.35i  1648  19.25  1649  19.36-1  1696  19.37-1  1697  spimt  1759  sbequi  1862