Theorem 19.35 1910

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 43	. . . 4 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
2	1	aleximi 1865	. . 3 ⊢ (∀𝑥𝜑 → (∃𝑥(𝜑 → 𝜓) → ∃𝑥𝜓))
3	2	com12 33	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
4		exnal 1860	. . . 4 ⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
5		pm2.21 124	. . . . 5 ⊢ (¬ 𝜑 → (𝜑 → 𝜓))
6	5	eximi 1868	. . . 4 ⊢ (∃𝑥 ¬ 𝜑 → ∃𝑥(𝜑 → 𝜓))
7	4, 6	sylbir 238	. . 3 ⊢ (¬ ∀𝑥𝜑 → ∃𝑥(𝜑 → 𝜓))
8		exa1 1871	. . 3 ⊢ (∃𝑥𝜓 → ∃𝑥(𝜑 → 𝜓))
9	7, 8	ja 188	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑 → 𝜓))
10	3, 9	impbii 212	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))

This proof depends on syntax axioms:  ¬ wn 3   → 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

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

This theorem is used by:  19.35i  1911  19.35ri  1912  19.25  1913  19.43  1915  nfimd  1927  19.37imv  1980  speimfwALT  1997  19.39  2023  19.24  2024  19.36v  2026  19.37v  2030  19.36  2269  19.37  2271  spimt  2420  grothprim  10834  bj-nnfim  37434  bj-spimt2  37477  bj-spimtv  37486