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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege56b | Structured version Visualization version GIF version |
Description: Lemma for frege57b 40252. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege56b | ⊢ ((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (𝑦 = 𝑥 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege55b 40250 | . 2 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
2 | frege9 40165 | . 2 ⊢ ((𝑦 = 𝑥 → 𝑥 = 𝑦) → ((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (𝑦 = 𝑥 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (𝑦 = 𝑥 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-12 2177 ax-13 2390 ax-ext 2795 ax-frege1 40143 ax-frege2 40144 ax-frege8 40162 ax-frege52c 40241 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-sbc 3775 |
This theorem is referenced by: frege57b 40252 |
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