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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege68c | Structured version Visualization version GIF version |
Description: Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege59c.a | ⊢ 𝐴 ∈ 𝐵 |
Ref | Expression |
---|---|
frege68c | ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege57aid 41046 | . 2 ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) | |
2 | frege59c.a | . . 3 ⊢ 𝐴 ∈ 𝐵 | |
3 | 2 | frege67c 41104 | . 2 ⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑))) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1540 ∈ wcel 2114 [wsbc 3680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 ax-frege1 40964 ax-frege2 40965 ax-frege8 40983 ax-frege52a 41031 ax-frege58b 41075 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ifp 1063 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-sbc 3681 |
This theorem is referenced by: frege70 41107 frege77 41114 frege116 41153 |
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