Theorem 19.21 1796

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "x is not free in φ." (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)

Hypothesis

Ref	Expression
19.21.1	⊢ Ⅎxφ

Assertion

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

Proof of Theorem 19.21

Step	Hyp	Ref	Expression
1		19.21.1	. 2 ⊢ Ⅎxφ
2		19.21t 1795	. 2 ⊢ (Ⅎxφ → (∀x(φ → ψ) ↔ (φ → ∀xψ)))
3	1, 2	ax-mp 5	1 ⊢ (∀x(φ → ψ) ↔ (φ → ∀xψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  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:  19.21h  1797  stdpc5  1798  nfim1OLD  1812  19.21-2  1864  nf3  1867  19.32  1875  19.21v  1890  19.12vv  1898  ax15  2021  eu2  2229  moanim  2260  r2alf  2650