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Mirrors > Home > MPE Home > Th. List > sb6OLD | Structured version Visualization version GIF version |
Description: Obsolete version of sb6 2089 as of 7-Jul-2023. Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. The implication "to the left", sb2vOLD 2093, also holds without a disjoint variable condition (sb2 2500). Theorem sb6f 2533 replaces the disjoint variable condition with a non-freeness hypothesis. Theorem sb4b 2495 replaces it with a distinctor antecedent. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb6OLD | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb4vOLD 2092 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
2 | sb2vOLD 2093 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑) | |
3 | 1, 2 | impbii 211 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-sb 2066 |
This theorem is referenced by: (None) |
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