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Theorem hba1-o 34686
Description: The setvar 𝑥 is not free in 𝑥𝜑. Example in Appendix in [Megill] p. 450 (p. 19 of the preprint). Also Lemma 22 of [Monk2] p. 114. (Contributed by NM, 24-Jan-1993.) (New usage is discouraged.)
Assertion
Ref Expression
hba1-o (∀𝑥𝜑 → ∀𝑥𝑥𝜑)

Proof of Theorem hba1-o
StepHypRef Expression
1 ax-c5 34672 . . 3 (∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝜑)
21con2i 134 . 2 (∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
3 ax10fromc7 34684 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑)
4 ax10fromc7 34684 . . . 4 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
54con1i 144 . . 3 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
65alimi 1888 . 2 (∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝑥𝜑)
72, 3, 63syl 18 1 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-c5 34672  ax-c4 34673  ax-c7 34674
This theorem is referenced by:  axc4i-o  34687  nfa1-o  34704  axc711toc7  34705  axc5c711toc7  34709  dvelimf-o  34718  ax12indalem  34734  ax12inda2ALT  34735  ax12inda  34737
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