Theorem 19.23v 1932

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 1575	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1547	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1396  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-gen 1498  ax-ie2 1543  ax-17 1575

This theorem is referenced by:  19.23vv  1933  2eu4  2174  gencbval  2863  euind  3004  reuind  3022  snssb  3827  unissb  3944  disjnim  4099  dftr2  4210  ssrelrel  4850  cotr  5144  dffun2  5362  fununi  5424  dff13  5941  acexmidlem2  6047