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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege64c | Structured version Visualization version GIF version |
Description: Lemma for frege65c 40272. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege59c.a | ⊢ 𝐴 ∈ 𝐵 |
Ref | Expression |
---|---|
frege64c | ⊢ (([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege59c.a | . . 3 ⊢ 𝐴 ∈ 𝐵 | |
2 | 1 | frege62c 40269 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥]𝜒)) |
3 | frege18 40162 | . 2 ⊢ (([𝐴 / 𝑥]𝜓 → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥]𝜒)) → (([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ (([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 ∈ wcel 2110 [wsbc 3771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2173 ax-ext 2793 ax-frege1 40134 ax-frege2 40135 ax-frege8 40153 ax-frege58b 40245 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-v 3496 df-sbc 3772 |
This theorem is referenced by: frege65c 40272 |
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