Theorem 19.36v 1944

Description: Version of 19.36 2162 with a disjoint variable condition instead of a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)

Assertion

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

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 19.36v

Step	Hyp	Ref	Expression
1		19.35 1840	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		19.9v 1938	. . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓)
3	2	imbi2i 328	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑 → 𝜓))
4	1, 3	bitri 267	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1505  ∃wex 1742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928

This theorem depends on definitions:  df-bi 199  df-ex 1743

This theorem is referenced by:  19.12vvv  1945  19.12vv  2281  ax13lem2  2305  axext2  2745  vtocl2OLD  3475  spcimdv  3505  bnj1090  31925  bj-spimvwt  33541  bj-spcimdv  33733  bj-spcimdvv  33734  19.36vv  40160