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Mirrors > Home > MPE Home > Th. List > axc11nlemOLD2 | Structured version Visualization version GIF version |
Description: Lemma for axc11n 2307. Change bound variable in an equality. Obsolete as of 29-Mar-2021. Use aev 1983 instead. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Restructure to ease either bundling, or reducing dependencies on axioms. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2047. (Revised by Wolf Lammen, 14-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc11nlemOLD2.1 | ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)) |
Ref | Expression |
---|---|
axc11nlemOLD2 | ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvaev 1979 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑧) | |
2 | equequ2 1953 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑥 ↔ 𝑦 = 𝑧)) | |
3 | 2 | biimprd 238 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑦 = 𝑧 → 𝑦 = 𝑥)) |
4 | 3 | al2imi 1743 | . . 3 ⊢ (∀𝑦 𝑥 = 𝑧 → (∀𝑦 𝑦 = 𝑧 → ∀𝑦 𝑦 = 𝑥)) |
5 | 1, 4 | syl5com 31 | . 2 ⊢ (∀𝑥 𝑥 = 𝑧 → (∀𝑦 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)) |
6 | spaev 1978 | . . . 4 ⊢ (∀𝑥 𝑥 = 𝑧 → 𝑥 = 𝑧) | |
7 | axc11nlemOLD2.1 | . . . 4 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧)) | |
8 | 6, 7 | syl5com 31 | . . 3 ⊢ (∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑥 = 𝑧)) |
9 | 8 | con1d 139 | . 2 ⊢ (∀𝑥 𝑥 = 𝑧 → (¬ ∀𝑦 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)) |
10 | 5, 9 | pm2.61d 170 | 1 ⊢ (∀𝑥 𝑥 = 𝑧 → ∀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: aevlemOLD 1989 axc11nOLDOLD 2309 |
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