Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > aevlemOLD | Structured version Visualization version GIF version |
Description: Old proof of aevlem 1981. Obsolete as of 29-Mar-2021. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2246, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
aevlemOLD | ⊢ (∀𝑧 𝑧 = 𝑤 → ∀𝑦 𝑦 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvaev 1979 | . 2 ⊢ (∀𝑧 𝑧 = 𝑤 → ∀𝑣 𝑣 = 𝑤) | |
2 | ax5d 1840 | . . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑣 → (𝑣 = 𝑤 → ∀𝑧 𝑣 = 𝑤)) | |
3 | 2 | axc11nlemOLD2 1988 | . 2 ⊢ (∀𝑣 𝑣 = 𝑤 → ∀𝑧 𝑧 = 𝑣) |
4 | cbvaev 1979 | . 2 ⊢ (∀𝑧 𝑧 = 𝑣 → ∀𝑥 𝑥 = 𝑣) | |
5 | ax5d 1840 | . . 3 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑣 → ∀𝑦 𝑥 = 𝑣)) | |
6 | 5 | axc11nlemOLD2 1988 | . 2 ⊢ (∀𝑥 𝑥 = 𝑣 → ∀𝑦 𝑦 = 𝑥) |
7 | 1, 3, 4, 6 | 4syl 19 | 1 ⊢ (∀𝑧 𝑧 = 𝑤 → ∀𝑦 𝑦 = 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |