Theorem 19.23v 1876

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 1519	. 2 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	19.23h 1491	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (∃𝑥𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1346  ∃wex 1485

This theorem was proved from axioms:  ax-mp 5  ax-gen 1442  ax-ie2 1487  ax-17 1519

This theorem is referenced by:  19.23vv  1877  2eu4  2112  gencbval  2778  euind  2917  reuind  2935  unissb  3826  disjnim  3980  dftr2  4089  ssrelrel  4711  cotr  4992  dffun2  5208  fununi  5266  dff13  5747  acexmidlem2  5850