Theorem 19.23v 1883

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 1526	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1498	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1351  ∃wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-gen 1449  ax-ie2 1494  ax-17 1526

This theorem is referenced by:  19.23vv  1884  2eu4  2119  gencbval  2786  euind  2925  reuind  2943  snssb  3726  unissb  3840  disjnim  3995  dftr2  4104  ssrelrel  4727  cotr  5011  dffun2  5227  fununi  5285  dff13  5769  acexmidlem2  5872