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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 34292 for more information. (Contributed by Anthony Hart, 11-Sep-2011.) |
Ref | Expression |
---|---|
subsym1 | ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fal 1557 | . . . . . . . . . 10 ⊢ ¬ ⊥ | |
2 | 1 | intnan 490 | . . . . . . . . 9 ⊢ ¬ (𝑥 = 𝑦 ∧ ⊥) |
3 | 2 | nex 1808 | . . . . . . . 8 ⊢ ¬ ∃𝑥(𝑥 = 𝑦 ∧ ⊥) |
4 | 3 | intnan 490 | . . . . . . 7 ⊢ ¬ ((𝑥 = 𝑦 → ⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ⊥)) |
5 | dfsb1 2484 | . . . . . . 7 ⊢ ([𝑦 / 𝑥]⊥ ↔ ((𝑥 = 𝑦 → ⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ⊥))) | |
6 | 4, 5 | mtbir 326 | . . . . . 6 ⊢ ¬ [𝑦 / 𝑥]⊥ |
7 | 6 | intnan 490 | . . . . 5 ⊢ ¬ (𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥) |
8 | 7 | nex 1808 | . . . 4 ⊢ ¬ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥) |
9 | 8 | intnan 490 | . . 3 ⊢ ¬ ((𝑥 = 𝑦 → [𝑦 / 𝑥]⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥)) |
10 | dfsb1 2484 | . . 3 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ ↔ ((𝑥 = 𝑦 → [𝑦 / 𝑥]⊥) ∧ ∃𝑥(𝑥 = 𝑦 ∧ [𝑦 / 𝑥]⊥))) | |
11 | 9, 10 | mtbir 326 | . 2 ⊢ ¬ [𝑦 / 𝑥][𝑦 / 𝑥]⊥ |
12 | 11 | pm2.21i 119 | 1 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ⊥wfal 1555 ∃wex 1787 [wsb 2072 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-10 2143 ax-12 2177 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 |
This theorem is referenced by: (None) |
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