Theorem 19.21 2136

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See 19.21v 1898 for a version requiring fewer axioms. See also 19.21h 2221. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1747 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 2135	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1505  Ⅎwnf 1746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-12 2106

This theorem depends on definitions:  df-bi 199  df-ex 1743  df-nf 1747

This theorem is referenced by:  stdpc5  2137  19.21-2  2138  19.32  2165  nf6  2217  19.21h  2221  19.12vv  2281  cbv1  2335  axc14  2400  r2alf  3173  19.12b  32564  bj-biexal2  33547  bj-bialal  33549  bj-cbv1v  33573  wl-dral1d  34210  mpobi123f  34881