Theorem 19.37v 1910

Description: Version of 19.37 2100 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 1897	. . 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 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  19.37iv  1911  eqvincg  3329  axrep5  4776  fvn0ssdmfun  6350  kmlem14  8985  kmlem15  8986  bnj132  30792  bnj1098  30854  bnj150  30946  bnj865  30993  bnj996  31025  bnj1021  31034  bnj1090  31047  bnj1176  31073  bj-axrep5  32792  cnvssco  37912  refimssco  37913  19.37vv  38584  pm11.61  38593  relopabVD  39137  rmoanim  41179