Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2485 with a distinct variable constraint. (Contributed by Wolf Lammen, 28-Sep-2018.) (Proof shortened by Wolf Lammen, 3-Oct-2023.) |
Ref | Expression |
---|---|
2ax6e | ⊢ ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aeveq 2052 | . . . . 5 ⊢ (∀𝑤 𝑤 = 𝑧 → 𝑧 = 𝑥) | |
2 | aeveq 2052 | . . . . 5 ⊢ (∀𝑤 𝑤 = 𝑧 → 𝑤 = 𝑦) | |
3 | 1, 2 | jca 512 | . . . 4 ⊢ (∀𝑤 𝑤 = 𝑧 → (𝑧 = 𝑥 ∧ 𝑤 = 𝑦)) |
4 | 3 | 19.8ad 2171 | . . 3 ⊢ (∀𝑤 𝑤 = 𝑧 → ∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)) |
5 | 4 | 19.8ad 2171 | . 2 ⊢ (∀𝑤 𝑤 = 𝑧 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)) |
6 | 2ax6elem 2485 | . 2 ⊢ (¬ ∀𝑤 𝑤 = 𝑧 → ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦)) | |
7 | 5, 6 | pm2.61i 183 | 1 ⊢ ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∀wal 1526 ∃wex 1771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-11 2151 ax-12 2167 ax-13 2381 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 |
This theorem is referenced by: 2sb5rf 2488 2sb6rf 2489 2sb6rfOLD 2490 |
Copyright terms: Public domain | W3C validator |