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Axiom ax-c9 33068
Description: Axiom of Quantifier Introduction. One of the equality and substitution axioms of predicate calculus with equality. Informally, it says that whenever 𝑧 is distinct from 𝑥 and 𝑦, and 𝑥 = 𝑦 is true, then 𝑥 = 𝑦 quantified with 𝑧 is also true. In other words, 𝑧 is irrelevant to the truth of 𝑥 = 𝑦. Axiom scheme C9' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases.

This axiom is obsolete and should no longer be used. It is proved above as theorem axc9 2194. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

Assertion
Ref Expression
ax-c9 (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))

Detailed syntax breakdown of Axiom ax-c9
StepHypRef Expression
1 vz . . . . 5 setvar 𝑧
2 vx . . . . 5 setvar 𝑥
31, 2weq 1824 . . . 4 wff 𝑧 = 𝑥
43, 1wal 1472 . . 3 wff 𝑧 𝑧 = 𝑥
54wn 3 . 2 wff ¬ ∀𝑧 𝑧 = 𝑥
6 vy . . . . . 6 setvar 𝑦
71, 6weq 1824 . . . . 5 wff 𝑧 = 𝑦
87, 1wal 1472 . . . 4 wff 𝑧 𝑧 = 𝑦
98wn 3 . . 3 wff ¬ ∀𝑧 𝑧 = 𝑦
102, 6weq 1824 . . . 4 wff 𝑥 = 𝑦
1110, 1wal 1472 . . . 4 wff 𝑧 𝑥 = 𝑦
1210, 11wi 4 . . 3 wff (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
139, 12wi 4 . 2 wff (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦))
145, 13wi 4 1 wff (¬ ∀𝑧 𝑧 = 𝑥 → (¬ ∀𝑧 𝑧 = 𝑦 → (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)))
Colors of variables: wff setvar class
This axiom is referenced by:  equid1  33077  hbae-o  33081  ax13fromc9  33084  hbequid  33087  equid1ALT  33103  dvelimf-o  33107  ax5eq  33110
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