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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege67c | Structured version Visualization version GIF version |
Description: Lemma for frege68c 40270. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege59c.a | ⊢ 𝐴 ∈ 𝐵 |
Ref | Expression |
---|---|
frege67c | ⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege59c.a | . . 3 ⊢ 𝐴 ∈ 𝐵 | |
2 | 1 | frege58c 40260 | . 2 ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑) |
3 | frege7 40147 | . 2 ⊢ ((∀𝑥𝜑 → [𝐴 / 𝑥]𝜑) → (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑)))) | |
4 | 2, 3 | ax-mp 5 | 1 ⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1531 ∈ wcel 2110 [wsbc 3771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-ext 2793 ax-frege1 40129 ax-frege2 40130 ax-frege58b 40240 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-clab 2800 df-cleq 2814 df-clel 2893 df-sbc 3772 |
This theorem is referenced by: frege68c 40270 |
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