Theorem 19.36iv 1892

Description: Inference associated with 19.36v 1891. Version of 19.36i 2086 with a dv condition. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)

Hypothesis

Ref	Expression
19.36iv.1	⊢ ∃𝑥(𝜑 → 𝜓)

Assertion

Ref	Expression
19.36iv	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 19.36iv

Step	Hyp	Ref	Expression
1		19.36iv.1	. 2 ⊢ ∃𝑥(𝜑 → 𝜓)
2		19.36v 1891	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
3	1, 2	mpbi 219	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by:  spimv  2245  vtocl  3232  vtocl2  3234  vtocl3  3235  zfcndext  9314  bj-spimvv  31908