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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege67b | Structured version Visualization version GIF version |
Description: Lemma for frege68b 41521. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege67b | ⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-frege58b 41509 | . 2 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
2 | frege7 41416 | . 2 ⊢ ((∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) → (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑)))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1537 [wsb 2067 |
This theorem was proved from axioms: ax-mp 5 ax-frege1 41398 ax-frege2 41399 ax-frege58b 41509 |
This theorem is referenced by: frege68b 41521 |
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