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Mirrors > Home > MPE Home > Th. List > 2ax6e | Structured version Visualization version GIF version |
Description: We can always find values matching 𝑥 and 𝑦, as long as they are represented by distinct variables. Version of 2ax6elem 2437 with a distinct variable constraint. (Contributed by Wolf Lammen, 28-Sep-2018.) |
Ref | Expression |
---|---|
2ax6e | ⊢ ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aeveq 1969 | . . . 4 ⊢ (∀𝑤 𝑤 = 𝑧 → 𝑧 = 𝑥) | |
2 | aeveq 1969 | . . . 4 ⊢ (∀𝑤 𝑤 = 𝑧 → 𝑤 = 𝑦) | |
3 | 1, 2 | jca 553 | . . 3 ⊢ (∀𝑤 𝑤 = 𝑧 → (𝑧 = 𝑥 ∧ 𝑤 = 𝑦)) |
4 | 19.8a 2039 | . . 3 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)) | |
5 | 19.8a 2039 | . . 3 ⊢ (∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)) | |
6 | 3, 4, 5 | 3syl 18 | . 2 ⊢ (∀𝑤 𝑤 = 𝑧 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)) |
7 | 2ax6elem 2437 | . 2 ⊢ (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)) | |
8 | 6, 7 | pm2.61i 175 | 1 ⊢ ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 |
This theorem is referenced by: 2sb5rf 2439 2sb6rf 2440 |
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