Theorem 19.37v 1999

Description: Version of 19.37 2240 with a disjoint variable 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 1879	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		19.3v 1984	. . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑)
3	2	imbi1i 349	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓))
4	1, 3	bitri 275	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969

This theorem depends on definitions:  df-bi 207  df-ex 1782

This theorem is referenced by:  spc3egv  3546  eqvincg  3591  rmoanim  3833  rmoanimALT  3834  axrep5  5221  fvn0ssdmfun  7021  kmlem14  10080  kmlem15  10081  bnj132  34888  bnj1098  34945  bnj150  35037  bnj865  35084  bnj996  35117  bnj1021  35127  bnj1090  35140  bnj1176  35166  sn-axrep5v  42675  cnvssco  44054  refimssco  44055  19.37vv  44833  pm11.61  44841  relopabVD  45348