Theorem 19.23 1801

Description: Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Hypothesis

Ref	Expression
19.23.1	⊢ Ⅎxψ

Assertion

Ref	Expression
19.23	⊢ (∀x(φ → ψ) ↔ (∃xφ → ψ))

Proof of Theorem 19.23

Step	Hyp	Ref	Expression
1		19.23.1	. 2 ⊢ Ⅎxψ
2		19.23t 1800	. 2 ⊢ (Ⅎxψ → (∀x(φ → ψ) ↔ (∃xφ → ψ)))
3	1, 2	ax-mp 5	1 ⊢ (∀x(φ → ψ) ↔ (∃xφ → ψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544

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:  19.23h  1802  exlimi  1803  exlimd  1806  nf2  1866  19.23v  1891  pm11.53  1893  ax10-16  2190  r19.3rz  3642  ralidm  3654