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Mirrors > Home > MPE Home > Th. List > Mathboxes > subsym1 | Structured version Visualization version GIF version |
Description: A symmetry with [𝑥 / 𝑦].
See negsym1 32416 for more information. (Contributed by Anthony Hart, 11-Sep-2011.) |
Ref | Expression |
---|---|
subsym1 | ⊢ ([𝑥 / 𝑦][𝑥 / 𝑦]⊥ → [𝑥 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fal 1490 | . . . . . . . . . 10 ⊢ ¬ ⊥ | |
2 | 1 | intnan 960 | . . . . . . . . 9 ⊢ ¬ (𝑦 = 𝑥 ∧ ⊥) |
3 | 2 | nex 1731 | . . . . . . . 8 ⊢ ¬ ∃𝑦(𝑦 = 𝑥 ∧ ⊥) |
4 | 3 | intnan 960 | . . . . . . 7 ⊢ ¬ ((𝑦 = 𝑥 → ⊥) ∧ ∃𝑦(𝑦 = 𝑥 ∧ ⊥)) |
5 | df-sb 1881 | . . . . . . 7 ⊢ ([𝑥 / 𝑦]⊥ ↔ ((𝑦 = 𝑥 → ⊥) ∧ ∃𝑦(𝑦 = 𝑥 ∧ ⊥))) | |
6 | 4, 5 | mtbir 313 | . . . . . 6 ⊢ ¬ [𝑥 / 𝑦]⊥ |
7 | 6 | intnan 960 | . . . . 5 ⊢ ¬ (𝑦 = 𝑥 ∧ [𝑥 / 𝑦]⊥) |
8 | 7 | nex 1731 | . . . 4 ⊢ ¬ ∃𝑦(𝑦 = 𝑥 ∧ [𝑥 / 𝑦]⊥) |
9 | 8 | intnan 960 | . . 3 ⊢ ¬ ((𝑦 = 𝑥 → [𝑥 / 𝑦]⊥) ∧ ∃𝑦(𝑦 = 𝑥 ∧ [𝑥 / 𝑦]⊥)) |
10 | df-sb 1881 | . . 3 ⊢ ([𝑥 / 𝑦][𝑥 / 𝑦]⊥ ↔ ((𝑦 = 𝑥 → [𝑥 / 𝑦]⊥) ∧ ∃𝑦(𝑦 = 𝑥 ∧ [𝑥 / 𝑦]⊥))) | |
11 | 9, 10 | mtbir 313 | . 2 ⊢ ¬ [𝑥 / 𝑦][𝑥 / 𝑦]⊥ |
12 | 11 | pm2.21i 116 | 1 ⊢ ([𝑥 / 𝑦][𝑥 / 𝑦]⊥ → [𝑥 / 𝑦]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ⊥wfal 1488 ∃wex 1704 [wsb 1880 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 |
This theorem depends on definitions: df-bi 197 df-an 386 df-tru 1486 df-fal 1489 df-ex 1705 df-sb 1881 |
This theorem is referenced by: (None) |
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