Theorem 19.37imv 1981

Description: One direction of 19.37v 2031 that can be proven without ax-6 2001. (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 1944	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		19.35 1911	. . 3 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
3	2	biimpi 219	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
4	1, 3	syl5 35	1 ⊢ (∃𝑥(𝜑 → 𝜓) → (𝜑 → ∃𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  19.37iv  1982  sbcg  3814  axprALT2  35798