Theorem 19.36v 1987

Description: Version of 19.36 2230 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 1877	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		19.9v 1983	. . 3 ⊢ (∃𝑥𝜓 ↔ 𝜓)
3	2	imbi2i 336	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑 → 𝜓))
4	1, 3	bitri 275	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by:  19.12vvv  1988  19.12vv  2349  ax13lem2  2381  axexte  2709  spcimdv  3593  bnj1090  34993  bj-spimvwt  36670  bj-spcimdv  36896  bj-spcimdvv  36897  19.36vv  44402