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Mirrors > Home > MPE Home > Th. List > ax12OLD | Structured version Visualization version GIF version |
Description: Obsolete proof of ax12 2304 as of 4-Jul-2021 . Rederivation of axiom ax-12 2047 from ax12v 2048, axc11r 2187, and other axioms. (Contributed by NM, 22-Jan-2007.) Proof uses contemporary axioms. (Revised by Wolf Lammen, 8-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax12OLD | ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 252 | . . . . 5 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 ↔ 𝜑)) | |
2 | 1 | dral1 2325 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝜑 ↔ ∀𝑦𝜑)) |
3 | ax-1 6 | . . . . 5 ⊢ (𝜑 → (𝑥 = 𝑦 → 𝜑)) | |
4 | 3 | alimi 1739 | . . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
5 | 2, 4 | syl6bir 244 | . . 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 | sp 2053 | . . 3 ⊢ (∀𝑦𝜑 → 𝜑) | |
8 | axc15 2303 | . . 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 176 | 1 ⊢ (𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 |
This theorem is referenced by: (None) |
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