Theorem 19.36imv 1946

Description: One direction of 19.36v 1994 that can be proven without ax-6 1970. (Contributed by Rohan Ridenour, 16-Apr-2022.)

Assertion

Ref	Expression
19.36imv	⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 19.36imv

Step	Hyp	Ref	Expression
1		19.35 1878	. . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2	1	biimpi 218	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
3		ax5e 1913	. 2 ⊢ (∃𝑥𝜓 → 𝜓)
4	2, 3	syl6 35	1 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 209  df-ex 1781

This theorem is referenced by:  19.36iv  1947