Theorem 19.37v 1997

Description: Version of 19.37 2233 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 1877	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		19.3v 1982	. . 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 1538  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by:  spc3egv  3566  eqvincg  3611  rmoanim  3854  rmoanimALT  3855  axrep5  5237  fvn0ssdmfun  7028  kmlem14  10093  kmlem15  10094  bnj132  34689  bnj1098  34746  bnj150  34839  bnj865  34886  bnj996  34919  bnj1021  34929  bnj1090  34942  bnj1176  34968  sn-axrep5v  42177  cnvssco  43568  refimssco  43569  19.37vv  44347  pm11.61  44355  relopabVD  44863