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Theorem 19.21 2242
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See 19.21v 1968 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 1813 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
Hypothesis
Ref Expression
19.21.1 𝑥𝜑
Assertion
Ref Expression
19.21 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Proof of Theorem 19.21
StepHypRef Expression
1 19.21.1 . 2 𝑥𝜑
2 19.21t 2241 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
31, 2ax-mp 5 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  wi 4  wb 209  wal 1567  wnf 1812
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-12 2212
This proof depends on definitions:  df-bi 210  df-ex 1809  df-nf 1813
This theorem is used by:  stdpc5  2243  19.21-2  2244  19.32  2268  nf6  2317  19.21h  2321  sbrim  2338  cbv1v  2367  19.12vv  2378  cbv1  2433  axc14  2494  r2alf  3285  19.12b  36299  bj-biexal2  37359  bj-bialal  37361  wl-dral1d  38214  mpobi123f  38839
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