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Theorem 19.21 2205
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". See 19.21v 1937 for a version requiring fewer axioms. See also 19.21h 2286. (Contributed by NM, 14-May-1993.) (Revised by Mario Carneiro, 24-Sep-2016.) df-nf 1781 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 2204 . 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 206  wal 1535  wnf 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-ex 1777  df-nf 1781
This theorem is referenced by:  stdpc5  2206  19.21-2  2207  19.32  2231  nf6  2282  19.21h  2286  sbrim  2303  cbv1v  2337  19.12vv  2348  cbv1  2405  axc14  2466  r2alf  3279  19.12b  35783  bj-biexal2  36689  bj-bialal  36691  wl-dral1d  37512  mpobi123f  38149
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