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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax-c11 | Structured version Visualization version GIF version |
Description: Axiom ax-c11 32692 was the original version of ax-c11n 32693 ("n" for "new"),
before it was discovered (in May 2008) that the shorter ax-c11n 32693 could
replace it. It appears as Axiom scheme C11' in [Megill] p. 448 (p. 16 of
the preprint).
This axiom is obsolete and should no longer be used. It is proved above as theorem axc11 2197. (Contributed by NM, 10-May-1993.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax-c11 | ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vx | . . . 4 setvar 𝑥 | |
2 | vy | . . . 4 setvar 𝑦 | |
3 | 1, 2 | weq 1822 | . . 3 wff 𝑥 = 𝑦 |
4 | 3, 1 | wal 1466 | . 2 wff ∀𝑥 𝑥 = 𝑦 |
5 | wph | . . . 4 wff 𝜑 | |
6 | 5, 1 | wal 1466 | . . 3 wff ∀𝑥𝜑 |
7 | 5, 2 | wal 1466 | . . 3 wff ∀𝑦𝜑 |
8 | 6, 7 | wi 4 | . 2 wff (∀𝑥𝜑 → ∀𝑦𝜑) |
9 | 4, 8 | wi 4 | 1 wff (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
Colors of variables: wff setvar class |
This axiom is referenced by: aecom-o 32704 hbae-o 32706 dral1-o 32707 axc11nfromc11 32730 dvelimf-o 32733 |
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