Theorem 19.23v 1856

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 1507	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1475	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1330  ∃wex 1469

This theorem was proved from axioms:  ax-mp 5  ax-gen 1426  ax-ie2 1471  ax-17 1507

This theorem is referenced by:  19.23vv  1857  2eu4  2093  gencbval  2737  euind  2875  reuind  2893  unissb  3774  disjnim  3928  dftr2  4036  ssrelrel  4647  cotr  4928  dffun2  5141  fununi  5199  dff13  5677  acexmidlem2  5779