Theorem 19.23v 1871

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 1514	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1486	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341  ∃wex 1480

This theorem was proved from axioms:  ax-mp 5  ax-gen 1437  ax-ie2 1482  ax-17 1514

This theorem is referenced by:  19.23vv  1872  2eu4  2107  gencbval  2774  euind  2913  reuind  2931  unissb  3819  disjnim  3973  dftr2  4082  ssrelrel  4704  cotr  4985  dffun2  5198  fununi  5256  dff13  5736  acexmidlem2  5839