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  3572  eqvincg  3617  rmoanim  3860  rmoanimALT  3861  axrep5  5245  fvn0ssdmfun  7049  kmlem14  10124  kmlem15  10125  bnj132  34723  bnj1098  34780  bnj150  34873  bnj865  34920  bnj996  34953  bnj1021  34963  bnj1090  34976  bnj1176  35002  sn-axrep5v  42211  cnvssco  43602  refimssco  43603  19.37vv  44381  pm11.61  44389  relopabVD  44897