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| Mirrors > Home > MPE Home > Th. List > Mathboxes > frege58bcor | Structured version Visualization version GIF version | ||
| Description: Lemma for frege59b 43917. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) | 
| Ref | Expression | 
|---|---|
| frege58bcor | ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-frege58b 43914 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → [𝑦 / 𝑥](𝜑 → 𝜓)) | |
| 2 | sbim 2303 | . 2 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | |
| 3 | 1, 2 | sylib 218 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∀wal 1538 [wsb 2064 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-12 2177 ax-frege58b 43914 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-sb 2065 | 
| This theorem is referenced by: frege59b 43917 frege62b 43920 | 
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