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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-ax12v3ALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of bj-ax12v3 36686. Uses axc11r 2371 and axc15 2427 instead of ax-12 2177. (Contributed by BJ, 6-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bj-ax12v3ALT | ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-5 1910 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) | |
| 2 | axc11r 2371 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑦𝜑 → ∀𝑥𝜑)) | |
| 3 | ala1 1813 | . . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
| 4 | 1, 2, 3 | syl56 36 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
| 5 | 4 | a1d 25 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) |
| 6 | axc15 2427 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))) | |
| 7 | 5, 6 | pm2.61i 182 | 1 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 ax-13 2377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 |
| This theorem is referenced by: (None) |
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