![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
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 39022 | . 2 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) | |
2 | sbsbc 3666 | . 2 ⊢ ([𝑥 / 𝑧]𝜑 ↔ [𝑥 / 𝑧]𝜑) | |
3 | sbsbc 3666 | . 2 ⊢ ([𝑦 / 𝑧]𝜑 ↔ [𝑦 / 𝑧]𝜑) | |
4 | 1, 2, 3 | 3imtr4g 288 | 1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2069 [wsbc 3662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-ext 2803 ax-frege52c 39022 |
This theorem depends on definitions: df-bi 199 df-an 387 df-ex 1881 df-clab 2812 df-cleq 2818 df-clel 2821 df-sbc 3663 |
This theorem is referenced by: frege53b 39024 frege57b 39033 |
Copyright terms: Public domain | W3C validator |