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Mirrors > Home > MPE Home > Th. List > Mathboxes > equid1ALT | Structured version Visualization version GIF version |
Description: Alternate proof of equid 1939 and equid1 34184 from older axioms ax-c7 34170, ax-c10 34171 and ax-c9 34175. (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 34175 | . . . . 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 1739 | . . 3 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → ∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥)) |
4 | ax-c10 34171 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑥 → ∀𝑥 𝑥 = 𝑥) → 𝑥 = 𝑥) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → 𝑥 = 𝑥) |
6 | ax-c7 34170 | . 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑥 → 𝑥 = 𝑥) | |
7 | 5, 6 | pm2.61i 176 | 1 ⊢ 𝑥 = 𝑥 |
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-c7 34170 ax-c10 34171 ax-c9 34175 |
This theorem is referenced by: (None) |
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