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| Mirrors > Home > MPE Home > Th. List > Mathboxes > equid1ALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of equid 2010 and equid1 38901 from older axioms ax-c7 38887, ax-c10 38888 and ax-c9 38892. (Contributed by NM, 10-Jan-1993.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| equid1ALT | ⊢ 𝑥 = 𝑥 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-c9 38892 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑥 → (¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥))) | |
| 2 | 1 | pm2.43i 52 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑥 → (𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) | 
| 3 | 2 | alimi 1810 | . . 3 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) | 
| 4 | ax-c10 38888 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) → 𝑥 = 𝑥) | |
| 5 | 3, 4 | syl 17 | . 2 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → 𝑥 = 𝑥) | 
| 6 | ax-c7 38887 | . 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → 𝑥 = 𝑥) | |
| 7 | 5, 6 | pm2.61i 182 | 1 ⊢ 𝑥 = 𝑥 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∀wal 1537 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-c7 38887 ax-c10 38888 ax-c9 38892 | 
| This theorem is referenced by: (None) | 
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