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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege60c | Structured version Visualization version GIF version |
Description: Swap antecedents of frege58c 38215. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege59c.a | ⊢ 𝐴 ∈ 𝐵 |
Ref | Expression |
---|---|
frege60c | ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜓 → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege59c.a | . . . 4 ⊢ 𝐴 ∈ 𝐵 | |
2 | 1 | frege58c 38215 | . . 3 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))) |
3 | sbcim1 3482 | . . . 4 ⊢ ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))) | |
4 | sbcim1 3482 | . . . 4 ⊢ ([𝐴 / 𝑥](𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒)) | |
5 | 3, 4 | syl6 35 | . . 3 ⊢ ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))) |
6 | 2, 5 | syl 17 | . 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))) |
7 | frege12 38107 | . 2 ⊢ ((∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → ([𝐴 / 𝑥]𝜓 → [𝐴 / 𝑥]𝜒))) → (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜓 → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))) | |
8 | 6, 7 | ax-mp 5 | 1 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜓 → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 ∈ wcel 1990 [wsbc 3435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-12 2047 ax-13 2246 ax-ext 2602 ax-frege1 38084 ax-frege2 38085 ax-frege8 38103 ax-frege58b 38195 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-v 3202 df-sbc 3436 |
This theorem is referenced by: frege93 38250 |
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