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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege53b | Structured version Visualization version GIF version |
Description: Lemma for frege102 (via frege92 43622). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege53b | ⊢ ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege52b 43556 | . 2 ⊢ (𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑)) | |
2 | ax-frege8 43476 | . 2 ⊢ ((𝑦 = 𝑧 → ([𝑦 / 𝑥]𝜑 → [𝑧 / 𝑥]𝜑)) → ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2060 |
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 2697 ax-frege8 43476 ax-frege52c 43555 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1775 df-clab 2704 df-cleq 2718 df-clel 2803 df-sbc 3777 |
This theorem is referenced by: frege55lem2b 43563 |
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