Theorem 19.23v 1931

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 1574	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1546	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1395  ∃wex 1540

This theorem was proved from axioms:  ax-mp 5  ax-gen 1497  ax-ie2 1542  ax-17 1574

This theorem is referenced by:  19.23vv  1932  2eu4  2173  gencbval  2852  euind  2993  reuind  3011  snssb  3806  unissb  3923  disjnim  4078  dftr2  4189  ssrelrel  4826  cotr  5118  dffun2  5336  fununi  5398  dff13  5908  acexmidlem2  6014