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Theorem 19.21 2209
 Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See 19.21v 1941 for a version requiring fewer axioms. See also 19.21h 2297. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1786 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 2208 . 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 209  ∀wal 1536  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-12 2179 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-nf 1786 This theorem is referenced by:  stdpc5  2210  19.21-2  2211  19.32  2237  nf6  2293  19.21h  2297  sbrimv  2316  cbv1v  2358  19.12vv  2370  cbv1  2424  axc14  2488  r2alf  3216  19.12b  33103  bj-biexal2  34097  bj-bialal  34099  wl-dral1d  34881  mpobi123f  35545
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