Theorem 19.37imv 1954

Description: One direction of 19.37v 1998 that can be proven without ax-6 1974. (Contributed by Rohan Ridenour, 16-Apr-2022.)

Assertion

Ref	Expression
19.37imv	⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.37imv

Step	Hyp	Ref	Expression
1		ax-5 1916	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		19.35 1883	. . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
3	2	biimpi 215	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
4	1, 3	syl5 34	1 ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1539  ∃wex 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916

This theorem depends on definitions:  df-bi 206  df-ex 1786

This theorem is referenced by:  19.37iv  1955  sbcg  3799