Theorem 19.37v 1912

Description: Version of 19.37 2103 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)

Assertion

Ref	Expression
19.37v	⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.37v

Step	Hyp	Ref	Expression
1		19.35 1805	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		19.3v 1899	. . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑)
3	2	imbi1i 339	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓))
4	1, 3	bitri 264	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1478  ∃wex 1701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890

This theorem depends on definitions:  df-bi 197  df-ex 1702

This theorem is referenced by:  19.37iv  1913  axrep5  4741  fvn0ssdmfun  6307  kmlem14  8930  kmlem15  8931  eqvincg  29154  bnj132  30492  bnj1098  30554  bnj150  30646  bnj865  30693  bnj996  30725  bnj1021  30734  bnj1090  30747  bnj1176  30773  bj-axrep5  32427  cnvssco  37379  refimssco  37380  19.37vv  38052  pm11.61  38061  relopabVD  38606  rmoanim  40470