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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege60b | Structured version Visualization version GIF version |
Description: Swap antecedents of ax-frege58b 41186. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege60b | ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-frege58b 41186 | . . 3 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → [𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒))) | |
2 | sbim 2304 | . . . 4 ⊢ ([𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → 𝜒))) | |
3 | sbim 2304 | . . . . 5 ⊢ ([𝑦 / 𝑥](𝜓 → 𝜒) ↔ ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)) | |
4 | 3 | imbi2i 339 | . . . 4 ⊢ (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥](𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))) |
5 | 2, 4 | bitri 278 | . . 3 ⊢ ([𝑦 / 𝑥](𝜑 → (𝜓 → 𝜒)) ↔ ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))) |
6 | 1, 5 | sylib 221 | . 2 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))) |
7 | frege12 41098 | . 2 ⊢ ((∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒))) → (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))) | |
8 | 6, 7 | ax-mp 5 | 1 ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 [wsb 2070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-12 2175 ax-frege1 41075 ax-frege2 41076 ax-frege8 41094 ax-frege58b 41186 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 df-sb 2071 |
This theorem is referenced by: (None) |
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