Theorem 19.23v 1891

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		nfv 1619	. 2 ⊢ Ⅎxψ
2	1	19.23 1801	1 ⊢ (∀x(φ → ψ) ↔ (∃xφ → ψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-ex 1542  df-nf 1545

This theorem is referenced by:  19.23vv  1892  2eu4  2287  euind  3024  reuind  3040  r19.3rzv  3644  unissb  3922  eqpw1  4163  pw111  4171  insklem  4305  cotr  5027  dffun2  5120  fununi  5161  dff13  5472  clos1induct  5881