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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege58c | Structured version Visualization version GIF version |
Description: Principle related to sp 2181. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
frege58c.a | ⊢ 𝐴 ∈ 𝐵 |
Ref | Expression |
---|---|
frege58c | ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege58c.a | . 2 ⊢ 𝐴 ∈ 𝐵 | |
2 | ax-frege58b 43891 | . . . . 5 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
3 | sbsbc 3795 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
4 | 2, 3 | sylib 218 | . . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) |
5 | dfsbcq 3793 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
6 | 4, 5 | imbitrid 244 | . . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) |
7 | 6 | vtocleg 3553 | . 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) |
8 | 1, 7 | ax-mp 5 | 1 ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 = wceq 1537 [wsb 2062 ∈ wcel 2106 [wsbc 3791 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-frege58b 43891 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-sbc 3792 |
This theorem is referenced by: frege59c 43912 frege60c 43913 frege61c 43914 frege62c 43915 frege67c 43920 frege72 43925 frege118 43971 frege120 43973 |
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