Theorem 19.23v 1863

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 1506	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1478	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1333  ∃wex 1472

This theorem was proved from axioms:  ax-mp 5  ax-gen 1429  ax-ie2 1474  ax-17 1506

This theorem is referenced by:  19.23vv  1864  2eu4  2099  gencbval  2760  euind  2899  reuind  2917  unissb  3802  disjnim  3956  dftr2  4064  ssrelrel  4685  cotr  4966  dffun2  5179  fununi  5237  dff13  5715  acexmidlem2  5818