Theorem 19.35 1771

Description: Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)

Assertion

Ref	Expression
19.35	⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem 19.35

Step	Hyp	Ref	Expression
1		pm2.27 40	. . . 4 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
2	1	aleximi 1735	. . 3 ⊢ (∀𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓))
3	2	com12 32	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
4		exnal 1730	. . . 4 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
5		pm2.21 113	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
6	5	eximi 1738	. . . 4 ⊢ (∃𝑥 ¬ 𝜑 → ∃𝑥(𝜑 → 𝜓))
7	4, 6	sylbir 220	. . 3 ⊢ (¬ ∀𝑥𝜑 → ∃𝑥(𝜑 → 𝜓))
8		exa1 1742	. . 3 ⊢ (∃𝑥𝜓 → ∃𝑥(𝜑 → 𝜓))
9	7, 8	ja 168	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))
10	3, 9	impbii 194	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 191  ∀wal 1466  ∃wex 1692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711

This theorem depends on definitions:  df-bi 192  df-ex 1693

This theorem is referenced by:  19.35i  1772  19.35ri  1773  19.25  1774  19.43  1776  speimfwALT  1825  19.39  1847  19.24  1848  19.36v  1852  19.37v  1858  19.36  2096  19.37  2098  spimt  2144  grothprim  9344  bj-nalnaleximiOLD  31404  bj-spimt2  31497  bj-spimtv  31506  bj-snsetex  31743