Theorem 19.23v 1894

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 1537	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1509	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362  ∃wex 1503

This theorem was proved from axioms:  ax-mp 5  ax-gen 1460  ax-ie2 1505  ax-17 1537

This theorem is referenced by:  19.23vv  1895  2eu4  2135  gencbval  2808  euind  2947  reuind  2965  snssb  3751  unissb  3865  disjnim  4020  dftr2  4129  ssrelrel  4759  cotr  5047  dffun2  5264  fununi  5322  dff13  5811  acexmidlem2  5915