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Mirrors > Home > MPE Home > Th. List > Mathboxes > frege58c | Structured version Visualization version GIF version |
Description: Principle related to sp 2171. 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 43380 | . . . . 5 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
3 | sbsbc 3782 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
4 | 2, 3 | sylib 217 | . . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) |
5 | dfsbcq 3780 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
6 | 4, 5 | imbitrid 243 | . . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) |
7 | 6 | vtocleg 3541 | . 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 1531 = wceq 1533 [wsb 2059 ∈ wcel 2098 [wsbc 3778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-frege58b 43380 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-sbc 3779 |
This theorem is referenced by: frege59c 43401 frege60c 43402 frege61c 43403 frege62c 43404 frege67c 43409 frege72 43414 frege118 43460 frege120 43462 |
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