Theorem 19.23v 1855

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 1474	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1329  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-gen 1425  ax-ie2 1470  ax-17 1506

This theorem is referenced by:  19.23vv  1856  2eu4  2090  gencbval  2729  euind  2866  reuind  2884  unissb  3761  disjnim  3915  dftr2  4023  ssrelrel  4634  cotr  4915  dffun2  5128  fununi  5186  dff13  5662  acexmidlem2  5764