Theorem 19.23v 1905

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 1548	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1520	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1370  ∃wex 1514

This theorem was proved from axioms:  ax-mp 5  ax-gen 1471  ax-ie2 1516  ax-17 1548

This theorem is referenced by:  19.23vv  1906  2eu4  2146  gencbval  2820  euind  2959  reuind  2977  snssb  3765  unissb  3879  disjnim  4034  dftr2  4143  ssrelrel  4773  cotr  5061  dffun2  5278  fununi  5336  dff13  5827  acexmidlem2  5931