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Axiom ax-c9 34648
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This axiom is obsolete and should no longer be used.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 2435. (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 2028 . . . 4 wff 𝑧 = 𝑥
43, 1wal 1618 . . 3 wff 𝑧 𝑧 = 𝑥
54wn 3 . 2 wff ¬ ∀𝑧 𝑧 = 𝑥
6 vy . . . . . 6 setvar 𝑦
71, 6weq 2028 . . . . 5 wff 𝑧 = 𝑦
87, 1wal 1618 . . . 4 wff 𝑧 𝑧 = 𝑦
98wn 3 . . 3 wff ¬ ∀𝑧 𝑧 = 𝑦
102, 6weq 2028 . . . 4 wff 𝑥 = 𝑦
1110, 1wal 1618 . . . 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  34657  hbae-o  34661  ax13fromc9  34664  hbequid  34667  equid1ALT  34683  dvelimf-o  34687  ax5eq  34690
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