Theorem 19.37v 1966

Description: Version of 19.37 2138 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 1845	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		19.3v 1954	. . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑)
3	2	imbi1i 338	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓))
4	1, 3	bitri 264	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1521  ∃wex 1744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945

This theorem depends on definitions:  df-bi 197  df-ex 1745

This theorem is referenced by:  19.37ivOLD  1967  eqvincg  3360  axrep5  4809  fvn0ssdmfun  6390  kmlem14  9023  kmlem15  9024  bnj132  30920  bnj1098  30980  bnj150  31072  bnj865  31119  bnj996  31151  bnj1021  31160  bnj1090  31173  bnj1176  31199  bj-axrep5  32917  cnvssco  38229  refimssco  38230  19.37vv  38901  pm11.61  38910  relopabVD  39451  rmoanim  41500