Theorem 19.23t 1800

Description: Closed form of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 7-Nov-2005.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)

Assertion

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

Proof of Theorem 19.23t

Step	Hyp	Ref	Expression
1		exim 1575	. . 3 ⊢ (∀x(φ → ψ) → (∃xφ → ∃xψ))
2		19.9t 1779	. . . 4 ⊢ (Ⅎxψ → (∃xψ ↔ ψ))
3	2	biimpd 198	. . 3 ⊢ (Ⅎxψ → (∃xψ → ψ))
4	1, 3	syl9r 67	. 2 ⊢ (Ⅎxψ → (∀x(φ → ψ) → (∃xφ → ψ)))
5		nfr 1761	. . . 4 ⊢ (Ⅎxψ → (ψ → ∀xψ))
6	5	imim2d 48	. . 3 ⊢ (Ⅎxψ → ((∃xφ → ψ) → (∃xφ → ∀xψ)))
7		19.38 1794	. . 3 ⊢ ((∃xφ → ∀xψ) → ∀x(φ → ψ))
8	6, 7	syl6 29	. 2 ⊢ (Ⅎxψ → ((∃xφ → ψ) → ∀x(φ → ψ)))
9	4, 8	impbid 183	1 ⊢ (Ⅎxψ → (∀x(φ → ψ) ↔ (∃xφ → ψ)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544

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

This theorem is used by:  19.23  1801  sbft  2025  axie2  2329  r19.23t  2729  ceqsalt  2882  vtoclgft  2906  sbciegft  3077