Theorem 19.21t 1795

Description: Closed form of Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 27-May-1997.) (Revised by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 3-Jan-2018.)

Assertion

Ref	Expression
19.21t	⊢ (Ⅎxφ → (∀x(φ → ψ) ↔ (φ → ∀xψ)))

Proof of Theorem 19.21t

Step	Hyp	Ref	Expression
1		nfr 1761	. . 3 ⊢ (Ⅎxφ → (φ → ∀xφ))
2		ax-5 1557	. . 3 ⊢ (∀x(φ → ψ) → (∀xφ → ∀xψ))
3	1, 2	syl9 66	. 2 ⊢ (Ⅎxφ → (∀x(φ → ψ) → (φ → ∀xψ)))
4		19.9t 1779	. . . 4 ⊢ (Ⅎxφ → (∃xφ ↔ φ))
5	4	imbi1d 308	. . 3 ⊢ (Ⅎxφ → ((∃xφ → ∀xψ) ↔ (φ → ∀xψ)))
6		19.38 1794	. . 3 ⊢ ((∃xφ → ∀xψ) → ∀x(φ → ψ))
7	5, 6	syl6bir 220	. 2 ⊢ (Ⅎxφ → ((φ → ∀xψ) → ∀x(φ → ψ)))
8	3, 7	impbid 183	1 ⊢ (Ⅎxφ → (∀x(φ → ψ) ↔ (φ → ∀xψ)))

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:  19.21  1796  nfimd  1808  sbcom  2089  sbal2  2134  ax11indalem  2197  ax11inda2ALT  2198  r19.21t  2700  ceqsalt  2882  sbciegft  3077