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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-sbcom2d-lem2 | Structured version Visualization version GIF version |
Description: Lemma used to prove wl-sbcom2d 34799. (Contributed by Wolf Lammen, 10-Aug-2019.) (New usage is discouraged.) |
Ref | Expression |
---|---|
wl-sbcom2d-lem2 | ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 𝑦 = 𝑥) | |
2 | naev 2065 | . 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 𝑦 = 𝑣) | |
3 | naev 2065 | . 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑦 𝑦 = 𝑢) | |
4 | naev 2065 | . 2 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ¬ ∀𝑥 𝑥 = 𝑢) | |
5 | 1, 2, 3, 4 | wl-2sb6d 34796 | 1 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 |
This theorem is referenced by: wl-sbcom2d 34799 |
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