Theorem 19.23v 1804

Description: Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.)

Assertion

Ref	Expression
19.23v	⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Distinct variable group:   𝜓,𝑥

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 19.23v

Step	Hyp	Ref	Expression
1		ax-17 1459	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1427	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1282  ∃wex 1421

This theorem was proved from axioms:  ax-mp 7  ax-gen 1378  ax-ie2 1423  ax-17 1459

This theorem is referenced by:  19.23vv  1805  2eu4  2034  gencbval  2647  euind  2779  reuind  2795  unissb  3631  dftr2  3877  ssrelrel  4458  cotr  4726  dffun2  4932  fununi  4987  dff13  5428  acexmidlem2  5529