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| Description: Principle related to sp 2182. 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 43919 | . . . . 5 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | |
| 3 | sbsbc 3791 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) | |
| 4 | 2, 3 | sylib 218 | . . . 4 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | 
| 5 | dfsbcq 3789 | . . . 4 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) | |
| 6 | 4, 5 | imbitrid 244 | . . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)) | 
| 7 | 6 | vtocleg 3552 | . 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 1537 = wceq 1539 [wsb 2063 ∈ wcel 2107 [wsbc 3787 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-frege58b 43919 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-sbc 3788 | 
| This theorem is referenced by: frege59c 43940 frege60c 43941 frege61c 43942 frege62c 43943 frege67c 43948 frege72 43953 frege118 43999 frege120 44001 | 
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