Theorem 19.36imv 1974

Description: One direction of 19.36v 2022 that can be proven without ax-6 1996. (Contributed by Rohan Ridenour, 16-Apr-2022.) (Proof shortened by Wolf Lammen, 22-Sep-2024.)

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		pm2.27 43	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
2	1	aleximi 1861	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓))
3		ax5e 1941	. 2 ⊢ (∃𝑥𝜓 → 𝜓)
4	2, 3	syl6com 38	1 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939

This proof depends on definitions:  df-bi 210  df-ex 1809

This theorem is used by:  19.36iv  1975