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Theorem 19.21 2192
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See 19.21v 1934 for a version requiring fewer axioms. See also 19.21h 2275. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1778 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 2191 . 2 (Ⅎ𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓)))
31, 2ax-mp 5 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-ex 1774  df-nf 1778
This theorem is referenced by:  stdpc5  2193  19.21-2  2194  19.32  2218  nf6  2271  19.21h  2275  sbrim  2292  cbv1v  2324  19.12vv  2335  cbv1  2393  axc14  2454  r2alf  3270  19.12b  35268  bj-biexal2  36074  bj-bialal  36076  wl-dral1d  36890  mpobi123f  37520
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