Theorem 19.36v 2026

Description: Version of 19.36 2268 with a disjoint variable condition instead of a nonfreeness 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 1910	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		19.9v 2017	. . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓)
3	2	imbi2i 339	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑 → 𝜓))
4	1, 3	bitri 278	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  19.12vvv  2027  19.12vv  2378  ax13lem2  2407  axexte  2735  spcimdv  3550  bnj1090  35473  bj-spimvwt  37385  bj-spcimdv  37623  bj-spcimdvv  37624  19.36vv  45192