Theorem 19.21 2230

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." See 19.21v 2019 for a version requiring fewer axioms. See also 19.21h 2283. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1857 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 2228	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1628  Ⅎwnf 1855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-12 2202

This theorem depends on definitions:  df-bi 197  df-ex 1852  df-nf 1857

This theorem is referenced by:  stdpc5  2231  19.21-2  2233  19.32  2256  nf6  2279  19.21h  2283  19.12vv  2341  cbv1  2427  axc14  2518  r2alf  3086  19.12b  32037  bj-biexal2  33028  bj-bialal  33030  bj-cbv1v  33059  wl-dral1d  33646  mpt2bi123f  34296