Theorem 19.21 2244

Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See 19.21v 1972 for a version requiring fewer axioms. See also 19.21h 2321. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1817 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 2243	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
3	1, 2	ax-mp 5	1 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2213

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  stdpc5  2245  19.21-2  2246  19.32  2270  nf6  2317  19.21h  2321  sbrim  2338  cbv1v  2366  19.12vv  2377  cbv1  2432  axc14  2493  r2alf  3284  19.12b  36543  bj-biexal2  37588  bj-bialal  37590  wl-dral1d  38443  mpobi123f  39074