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Mirrors > Home > MPE Home > Th. List > Mathboxes > pm13.194 | Structured version Visualization version GIF version |
Description: Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) |
Ref | Expression |
---|---|
pm13.194 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm13.13a 42779 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑) | |
2 | sbsbc 3747 | . . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
3 | 1, 2 | sylibr 233 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → [𝑦 / 𝑥]𝜑) |
4 | simpl 484 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝜑) | |
5 | simpr 486 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦) | |
6 | 3, 4, 5 | 3jca 1129 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦)) |
7 | 3simpc 1151 | . 2 ⊢ (([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦) → (𝜑 ∧ 𝑥 = 𝑦)) | |
8 | 6, 7 | impbii 208 | 1 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝜑 ∧ 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∧ w3a 1088 [wsb 2068 [wsbc 3743 |
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 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-sbc 3744 |
This theorem is referenced by: (None) |
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