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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege60c | Structured version Visualization version GIF version |
Description: Swap antecedents of frege58c 40980. 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 40980 | . . 3 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → [𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒))) |
3 | sbcim1 3745 | . . . 4 ⊢ ([𝐴 / 𝑥](𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥](𝜓 → 𝜒))) | |
4 | sbcim1 3745 | . . . 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 40872 | . 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 1537 ∈ wcel 2112 [wsbc 3693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-12 2176 ax-ext 2730 ax-frege1 40849 ax-frege2 40850 ax-frege8 40868 ax-frege58b 40960 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-v 3409 df-sbc 3694 |
This theorem is referenced by: frege93 41015 |
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