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Theorem 19.21h 1465
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 1491 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 1362 . . 3 (∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
31, 2syl5 32 . 2 (∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
4 hba1 1449 . . . 4 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
51, 4hbim 1453 . . 3 ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓))
6 ax-4 1416 . . . 4 (∀𝑥𝜓𝜓)
76imim2i 12 . . 3 ((𝜑 → ∀𝑥𝜓) → (𝜑𝜓))
85, 7alrimih 1374 . 2 ((𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
93, 8impbii 121 1 (∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wal 1257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  hbim1  1478  nf3  1575  19.21v  1769
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