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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege68b | 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 |
---|---|
frege68b | ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege57aid 37186 | . 2 ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) | |
2 | frege67b 37226 | . 2 ⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∀wal 1473 [wsb 1867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-frege1 37104 ax-frege2 37105 ax-frege8 37123 ax-frege52a 37171 ax-frege58b 37215 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-ifp 1007 df-tru 1478 df-fal 1481 |
This theorem is referenced by: (None) |
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