Theorem 19.23v 1907

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 1550	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1522	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1371  ∃wex 1516

This theorem was proved from axioms:  ax-mp 5  ax-gen 1473  ax-ie2 1518  ax-17 1550

This theorem is referenced by:  19.23vv  1908  2eu4  2148  gencbval  2823  euind  2964  reuind  2982  snssb  3772  unissb  3886  disjnim  4041  dftr2  4152  ssrelrel  4783  cotr  5073  dffun2  5290  fununi  5351  dff13  5850  acexmidlem2  5954