Theorem 19.37v 2024

Description: Version of 19.37 2274 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 1904	. 2 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2		19.3v 2009	. . 3 ⊢ (∀𝑥𝜑 ↔ 𝜑)
3	2	imbi1i 352	. 2 ⊢ ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓))
4	1, 3	bitri 278	1 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1565  ∃wex 1806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994

This theorem depends on definitions:  df-bi 210  df-ex 1807

This theorem is referenced by:  spc3egv  3571  eqvincg  3616  rmoanim  3856  rmoanimALT  3857  axrep5  5250  fvn0ssdmfun  7070  kmlem14  10146  kmlem15  10147  bnj132  35059  bnj1098  35116  bnj150  35208  bnj865  35255  bnj996  35288  bnj1021  35298  bnj1090  35311  bnj1176  35337  sn-axrep5v  42877  cnvssco  44223  refimssco  44224  19.37vv  44986  pm11.61  44994  relopabVD  45500