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Axiom ax-c9 38362
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This axiom is obsolete and should no longer be used.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 2377. (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 1959 . . . 4 wff 𝑧 = 𝑥
43, 1wal 1532 . . 3 wff 𝑧 𝑧 = 𝑥
54wn 3 . 2 wff ¬ ∀𝑧 𝑧 = 𝑥
6 vy . . . . . 6 setvar 𝑦
71, 6weq 1959 . . . . 5 wff 𝑧 = 𝑦
87, 1wal 1532 . . . 4 wff 𝑧 𝑧 = 𝑦
98wn 3 . . 3 wff ¬ ∀𝑧 𝑧 = 𝑦
102, 6weq 1959 . . . 4 wff 𝑥 = 𝑦
1110, 1wal 1532 . . . 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  38371  hbae-o  38375  ax13fromc9  38378  hbequid  38381  equid1ALT  38397  dvelimf-o  38401  ax5eq  38404
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