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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 41202 | . 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) | |
2 | sbsbc 3713 | . 2 ⊢ ([𝑥 / 𝑧]𝜑 ↔ [𝑥 / 𝑧]𝜑) | |
3 | sbsbc 3713 | . 2 ⊢ ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑) | |
4 | 1, 2, 3 | 3imtr4g 299 | 1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2071 [wsbc 3709 |
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-8 2113 ax-9 2121 ax-ext 2709 ax-frege52c 41202 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-clab 2716 df-cleq 2730 df-clel 2817 df-sbc 3710 |
This theorem is referenced by: frege53b 41204 frege57b 41213 |
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