Theorem 19.23v 1763

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 1419	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1387	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  ∃wex 1381

This theorem was proved from axioms:  ax-mp 7  ax-gen 1338  ax-ie2 1383  ax-17 1419

This theorem is referenced by:  19.23vv  1764  2eu4  1993  gencbval  2602  euind  2728  reuind  2744  unissb  3610  dftr2  3856  ssrelrel  4440  cotr  4706  dffun2  4912  fununi  4967  dff13  5407  acexmidlem2  5509