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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-axc10 | Structured version Visualization version GIF version |
Description: Alternate (shorter) proof of axc10 2392. One can prove a version with DV (𝑥, 𝑦) without ax-13 2379, by using ax6ev 1972 instead of ax6e 2390. (Contributed by BJ, 31-Mar-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-axc10 | ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6e 2390 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | exim 1835 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑥𝜑)) | |
3 | 1, 2 | mpi 20 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ∃𝑥∀𝑥𝜑) |
4 | axc7e 2326 | . 2 ⊢ (∃𝑥∀𝑥𝜑 → 𝜑) | |
5 | 3, 4 | syl 17 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 ∃wex 1781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 |
This theorem is referenced by: (None) |
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