Theorem 19.37v 2030

Description: Version of 19.37 2271 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 1910	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		19.3v 2015	. . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑)
3	2	imbi1i 352	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓))
4	1, 3	bitri 278	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  spc3egv  3564  eqvincg  3609  rmoanim  3849  rmoanimALT  3850  axrep5  5248  fvn0ssdmfun  7073  kmlem14  10163  kmlem15  10164  bnj132  35182  bnj1098  35239  bnj150  35331  bnj865  35378  bnj996  35411  bnj1021  35421  bnj1090  35434  bnj1176  35460  sn-axrep5v  43048  cnvssco  44392  refimssco  44393  19.37vv  45155  pm11.61  45163  relopabVD  45669