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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax12fromc15 | Structured version Visualization version GIF version |
Description: Rederivation of Axiom ax-12 2164 from ax-c15 38285, ax-c11 38283 (used through
dral1-o 38300), and other older axioms. See Theorem axc15 2416 for the
derivation of ax-c15 38285 from ax-12 2164.
An open problem is whether we can prove this using ax-c11n 38284 instead of ax-c11 38283. This proof uses newer axioms ax-4 1804 and ax-6 1964, but since these are proved from the older axioms above, this is acceptable and lets us avoid having to reprove several earlier theorems to use ax-c4 38280 and ax-c10 38282. (Contributed by NM, 22-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax12fromc15 | ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 262 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜑)) | |
2 | 1 | dral1-o 38300 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜑)) |
3 | ax-1 6 | . . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → 𝜑)) | |
4 | 3 | alimi 1806 | . . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
5 | 2, 4 | syl6bir 254 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
6 | 5 | a1d 25 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
7 | ax-c5 38279 | . . 3 ⊢ (∀𝑦𝜑 → 𝜑) | |
8 | ax-c15 38285 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | |
9 | 7, 8 | syl7 74 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
10 | 6, 9 | pm2.61i 182 | 1 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1532 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-11 2147 ax-c5 38279 ax-c4 38280 ax-c7 38281 ax-c10 38282 ax-c11 38283 ax-c15 38285 ax-c9 38286 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 |
This theorem is referenced by: (None) |
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