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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm13.196a | Structured version Visualization version GIF version |
Description: Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.) |
Ref | Expression |
---|---|
pm13.196a | ⊢ (¬ 𝜑 ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 ≠ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbelx 2255 | . 2 ⊢ (¬ 𝜑 ↔ ∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥] ¬ 𝜑)) | |
2 | sbalex 2244 | . 2 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ [𝑦 / 𝑥] ¬ 𝜑) ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥] ¬ 𝜑)) | |
3 | sbn 2285 | . . . . 5 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑) | |
4 | 3 | imbi2i 339 | . . . 4 ⊢ ((𝑦 = 𝑥 → [𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑦 = 𝑥 → ¬ [𝑦 / 𝑥]𝜑)) |
5 | con2b 363 | . . . 4 ⊢ ((𝑦 = 𝑥 → ¬ [𝑦 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 → ¬ 𝑦 = 𝑥)) | |
6 | df-ne 2936 | . . . . . 6 ⊢ (𝑦 ≠ 𝑥 ↔ ¬ 𝑦 = 𝑥) | |
7 | 6 | bicomi 227 | . . . . 5 ⊢ (¬ 𝑦 = 𝑥 ↔ 𝑦 ≠ 𝑥) |
8 | 7 | imbi2i 339 | . . . 4 ⊢ (([𝑦 / 𝑥]𝜑 → ¬ 𝑦 = 𝑥) ↔ ([𝑦 / 𝑥]𝜑 → 𝑦 ≠ 𝑥)) |
9 | 4, 5, 8 | 3bitri 300 | . . 3 ⊢ ((𝑦 = 𝑥 → [𝑦 / 𝑥] ¬ 𝜑) ↔ ([𝑦 / 𝑥]𝜑 → 𝑦 ≠ 𝑥)) |
10 | 9 | albii 1826 | . 2 ⊢ (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥] ¬ 𝜑) ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 ≠ 𝑥)) |
11 | 1, 2, 10 | 3bitri 300 | 1 ⊢ (¬ 𝜑 ↔ ∀𝑦([𝑦 / 𝑥]𝜑 → 𝑦 ≠ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1540 ∃wex 1786 [wsb 2074 ≠ wne 2935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-10 2145 ax-12 2179 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ex 1787 df-nf 1791 df-sb 2075 df-ne 2936 |
This theorem is referenced by: (None) |
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