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Mirrors > Home > MPE Home > Th. List > dfsb3 | Structured version Visualization version GIF version |
Description: An alternate definition of proper substitution df-sb 2066 that uses only primitive connectives (no defined terms) on the right-hand side. Usage of this theorem is discouraged because it depends on ax-13 2386. (Contributed by NM, 6-Mar-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfsb3 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-or 844 | . 2 ⊢ (((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (¬ (𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
2 | dfsb2 2528 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 ∧ 𝜑) ∨ ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
3 | imnan 402 | . . 3 ⊢ ((𝑥 = 𝑦 → ¬ 𝜑) ↔ ¬ (𝑥 = 𝑦 ∧ 𝜑)) | |
4 | 3 | imbi1i 352 | . 2 ⊢ (((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (¬ (𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
5 | 1, 2, 4 | 3bitr4i 305 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∀wal 1531 [wsb 2065 |
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 1907 ax-6 1966 ax-7 2011 ax-10 2141 ax-12 2173 ax-13 2386 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1777 df-nf 1781 df-sb 2066 |
This theorem is referenced by: sbnOLD 2537 |
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