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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege53b | Structured version Visualization version GIF version |
Description: Lemma for frege102 (via frege92 38566). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege53b | ⊢ ([𝑥 / 𝑦]𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑦]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege52b 38500 | . 2 ⊢ (𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜑)) | |
2 | ax-frege8 38420 | . 2 ⊢ ((𝑥 = 𝑧 → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜑)) → ([𝑥 / 𝑦]𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑦]𝜑))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ([𝑥 / 𝑦]𝜑 → (𝑥 = 𝑧 → [𝑧 / 𝑦]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 1937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-ext 2631 ax-frege8 38420 ax-frege52c 38499 |
This theorem depends on definitions: df-bi 197 df-an 385 df-ex 1745 df-clab 2638 df-cleq 2644 df-clel 2647 df-sbc 3469 |
This theorem is referenced by: frege55lem2b 38507 |
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