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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege55lem1b | Structured version Visualization version GIF version |
Description: Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) |
Ref | Expression |
---|---|
frege55lem1b | ⊢ ((𝜑 → [𝑥 / 𝑦]𝑦 = 𝑧) → (𝜑 → 𝑥 = 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsb3 2105 | . . 3 ⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧) | |
2 | 1 | biimpi 215 | . 2 ⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 → 𝑥 = 𝑧) |
3 | 2 | imim2i 16 | 1 ⊢ ((𝜑 → [𝑥 / 𝑦]𝑦 = 𝑧) → (𝜑 → 𝑥 = 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1787 df-sb 2072 |
This theorem is referenced by: (None) |
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