Theorem 19.23v 1929

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 1572	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1544	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1393  ∃wex 1538

This theorem was proved from axioms:  ax-mp 5  ax-gen 1495  ax-ie2 1540  ax-17 1572

This theorem is referenced by:  19.23vv  1930  2eu4  2171  gencbval  2850  euind  2991  reuind  3009  snssb  3804  unissb  3921  disjnim  4076  dftr2  4187  ssrelrel  4824  cotr  5116  dffun2  5334  fununi  5395  dff13  5904  acexmidlem2  6010