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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege67c | Structured version Visualization version GIF version |
Description: Lemma for frege68c 43834. 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 43824 | . 2 ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑) |
3 | frege7 43711 | . 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 206 ∀wal 1535 ∈ wcel 2103 [wsbc 3798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 ax-frege1 43693 ax-frege2 43694 ax-frege58b 43804 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-sbc 3799 |
This theorem is referenced by: frege68c 43834 |
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