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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 2399. One can prove a version with DV (𝑥, 𝑦) without ax-13 2386, by using ax6ev 1968 instead of ax6e 2397. (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 2397 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
2 | exim 1830 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∃𝑥 𝑥 = 𝑦 → ∃𝑥∀𝑥𝜑)) | |
3 | 1, 2 | mpi 20 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ∃𝑥∀𝑥𝜑) |
4 | axc7e 2333 | . 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 1531 ∃wex 1776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-12 2172 ax-13 2386 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 |
This theorem is referenced by: (None) |
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