Theorem 19.21 2060

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑." See 19.21v 1854 for a version requiring fewer axioms. See also 19.21h 2105. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1700 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 2059	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 194  ∀wal 1472  Ⅎwnf 1698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2032

This theorem depends on definitions:  df-bi 195  df-ex 1695  df-nf 1700

This theorem is referenced by:  stdpc5  2061  19.21-2  2063  19.32  2086  nf6  2101  19.21h  2105  19.12vv  2166  cbv1  2253  axc14  2359  r2alf  2921  19.12b  30757  bj-biexal2  31690  bj-bialal  31692  bj-cbv1v  31722  wl-dral1d  32300  mpt2bi123f  32944