Theorem i19.39 1531

Description: Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1515, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.)

Hypothesis

Ref	Expression
i19.24.1	⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))

Assertion

Ref	Expression
i19.39	⊢ ((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))

Proof of Theorem i19.39

Step	Hyp	Ref	Expression
1		19.2 1529	. . 3 ⊢ (∀𝑥𝜑 → ∃𝑥𝜑)
2	1	imim1i 54	. 2 ⊢ ((∃𝑥𝜑 → ∃𝑥𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
3		i19.24.1	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))
4	2, 3	syl 14	1 ⊢ ((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1241  ∃wex 1381

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)