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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege64c | Structured version Visualization version GIF version |
Description: Lemma for frege65c 43134. 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 43131 | . 2 ⊢ ([𝐴 / 𝑥]𝜓 → (∀𝑥(𝜓 → 𝜒) → [𝐴 / 𝑥]𝜒)) |
3 | frege18 43024 | . 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 2098 [wsbc 3769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-frege1 42996 ax-frege2 42997 ax-frege8 43015 ax-frege58b 43107 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-sbc 3770 |
This theorem is referenced by: frege65c 43134 |
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