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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm13.193 | Structured version Visualization version GIF version |
Description: Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
Ref | Expression |
---|---|
pm13.193 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbequ12 2249 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
2 | 1 | pm5.32ri 579 | 1 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 [wsb 2070 |
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 2016 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-sb 2071 |
This theorem is referenced by: (None) |
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