Theorem 19.23v 1897

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 1540	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1512	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362  ∃wex 1506

This theorem was proved from axioms:  ax-mp 5  ax-gen 1463  ax-ie2 1508  ax-17 1540

This theorem is referenced by:  19.23vv  1898  2eu4  2138  gencbval  2812  euind  2951  reuind  2969  snssb  3756  unissb  3870  disjnim  4025  dftr2  4134  ssrelrel  4764  cotr  5052  dffun2  5269  fununi  5327  dff13  5818  acexmidlem2  5922