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Mirrors > Home > MPE Home > Th. List > Mathboxes > axc11-o | Structured version Visualization version GIF version |
Description: Show that ax-c11 33190 can be derived from ax-c11n 33191 and ax-12 2034. An open
problem is whether this theorem can be derived from ax-c11n 33191 and the
others when ax-12 2034 is replaced with ax-c15 33192 or ax12v 2035. See theorem
axc11nfromc11 33229 for the rederivation of ax-c11n 33191 from axc11 2302.
Normally, axc11 2302 should be used rather than ax-c11 33190 or axc11-o 33254, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc11-o | ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-c11n 33191 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥) | |
2 | ax12 2292 | . . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑))) | |
3 | 2 | equcoms 1934 | . . 3 ⊢ (𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑))) |
4 | 3 | sps-o 33211 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦(𝑦 = 𝑥 → 𝜑))) |
5 | pm2.27 41 | . . 3 ⊢ (𝑦 = 𝑥 → ((𝑦 = 𝑥 → 𝜑) → 𝜑)) | |
6 | 5 | al2imi 1733 | . 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∀𝑦(𝑦 = 𝑥 → 𝜑) → ∀𝑦𝜑)) |
7 | 1, 4, 6 | sylsyld 59 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 → ∀𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 ax-13 2234 ax-c5 33186 ax-c11n 33191 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 |
This theorem is referenced by: (None) |
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