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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-aetr | Structured version Visualization version GIF version |
Description: A transitive law for variable identifying expressions. (Contributed by Wolf Lammen, 30-Jun-2019.) |
Ref | Expression |
---|---|
wl-aetr | ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax7 2011 | . . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝑧 → 𝑦 = 𝑧)) | |
2 | 1 | al2imi 1809 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑥 𝑦 = 𝑧)) |
3 | axc11 2423 | . 2 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑦 = 𝑧 → ∀𝑦 𝑦 = 𝑧)) | |
4 | 2, 3 | syld 47 | 1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-12 2163 ax-13 2365 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-nf 1778 |
This theorem is referenced by: wl-ax11-lem1 36958 wl-ax11-lem3 36960 |
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