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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 35302 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 | sbv 2092 | . . 3 ⊢ ([𝑦 / 𝑥]⊥ ↔ ⊥) | |
2 | falim 1559 | . . 3 ⊢ (⊥ → 𝜑) | |
3 | 1, 2 | sylbi 216 | . 2 ⊢ ([𝑦 / 𝑥]⊥ → 𝜑) |
4 | 3 | sbimi 2078 | 1 ⊢ ([𝑦 / 𝑥][𝑦 / 𝑥]⊥ → [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊥wfal 1554 [wsb 2068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 |
This theorem depends on definitions: df-bi 206 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 |
This theorem is referenced by: (None) |
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