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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege52b | Structured version Visualization version GIF version |
Description: The case when the content of 𝑥 is identical with the content of 𝑦 and in which a proposition controlled by an element for which we substitute the content of 𝑥 is affirmed and the same proposition, this time where we substitute the content of 𝑦, is denied does not take place. In [𝑥 / 𝑧]𝜑, 𝑥 can also occur in other than the argument (𝑧) places. Hence 𝑥 may still be contained in [𝑦 / 𝑧]𝜑. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege52b | ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-frege52c 43318 | . 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) | |
2 | sbsbc 3780 | . 2 ⊢ ([𝑥 / 𝑧]𝜑 ↔ [𝑥 / 𝑧]𝜑) | |
3 | sbsbc 3780 | . 2 ⊢ ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑) | |
4 | 1, 2, 3 | 3imtr4g 296 | 1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2060 [wsbc 3776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-frege52c 43318 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-clab 2706 df-cleq 2720 df-clel 2806 df-sbc 3777 |
This theorem is referenced by: frege53b 43320 frege57b 43329 |
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