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Theorem 19.21h 1557
Description: Theorem 19.21 of [Margaris] p. 90. The hypothesis can be thought of as "𝑥 is not free in 𝜑". New proofs should use 19.21 1583 instead. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
Hypothesis
Ref Expression
19.21h.1 (𝜑 → ∀𝑥𝜑)
Assertion
Ref Expression
19.21h (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))

Proof of Theorem 19.21h
StepHypRef Expression
1 19.21h.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 alim 1457 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
31, 2syl5 32 . 2 (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
4 hba1 1540 . . . 4 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
51, 4hbim 1545 . . 3 ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))
6 ax-4 1510 . . . 4 (∀𝑥𝜓𝜓)
76imim2i 12 . . 3 ((𝜑 → ∀𝑥𝜓) → (𝜑𝜓))
85, 7alrimih 1469 . 2 ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
93, 8impbii 126 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-4 1510  ax-ial 1534  ax-i5r 1535
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  hbim1  1570  nf3  1669  19.21v  1873
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