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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege67c | Structured version Visualization version GIF version |
Description: Lemma for frege68c 40284. 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 40274 | . 2 ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑) |
3 | frege7 40161 | . 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 1535 ∈ wcel 2114 [wsbc 3774 |
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 2795 ax-frege1 40143 ax-frege2 40144 ax-frege58b 40254 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-clab 2802 df-cleq 2816 df-clel 2895 df-sbc 3775 |
This theorem is referenced by: frege68c 40284 |
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