Theorem 19.37v 1998

Description: Version of 19.37 2237 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 1878	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		19.3v 1983	. . 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 1539  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968

This theorem depends on definitions:  df-bi 207  df-ex 1781

This theorem is referenced by:  spc3egv  3554  eqvincg  3599  rmoanim  3841  rmoanimALT  3842  axrep5  5227  fvn0ssdmfun  7013  kmlem14  10062  kmlem15  10063  bnj132  34759  bnj1098  34816  bnj150  34909  bnj865  34956  bnj996  34989  bnj1021  34999  bnj1090  35012  bnj1176  35038  sn-axrep5v  42334  cnvssco  43723  refimssco  43724  19.37vv  44502  pm11.61  44510  relopabVD  45017