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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege56b | Structured version Visualization version GIF version |
Description: Lemma for frege57b 42151. 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 42149 | . 2 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦) | |
2 | frege9 42064 | . 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 2067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-13 2370 ax-ext 2707 ax-frege1 42042 ax-frege2 42043 ax-frege8 42061 ax-frege52c 42140 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-sbc 3739 |
This theorem is referenced by: frege57b 42151 |
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