Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
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 40238 | . 2 ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) | |
2 | frege59c.a | . . 3 ⊢ 𝐴 ∈ 𝐵 | |
3 | 2 | frege67c 40296 | . 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 208 ∀wal 1535 ∈ wcel 2114 [wsbc 3772 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 ax-frege1 40156 ax-frege2 40157 ax-frege8 40175 ax-frege52a 40223 ax-frege58b 40267 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-tru 1540 df-fal 1550 df-ex 1781 df-clab 2800 df-cleq 2814 df-clel 2893 df-sbc 3773 |
This theorem is referenced by: frege70 40299 frege77 40306 frege116 40345 |
Copyright terms: Public domain | W3C validator |