Theorem 19.21 2210

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See 19.21v 1940 for a version requiring fewer axioms. See also 19.21h 2289. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1785 changed. (Revised by Wolf Lammen, 18-Sep-2021.)

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 2209	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2180

This theorem depends on definitions:  df-bi 207  df-ex 1781  df-nf 1785

This theorem is referenced by:  stdpc5  2211  19.21-2  2212  19.32  2236  nf6  2285  19.21h  2289  sbrim  2306  cbv1v  2336  19.12vv  2347  cbv1  2402  axc14  2463  r2alf  3253  19.12b  35843  bj-biexal2  36750  bj-bialal  36752  wl-dral1d  37575  mpobi123f  38212