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Mirrors > Home > MPE Home > Th. List > sb5f | Structured version Visualization version GIF version |
Description: Equivalence for substitution when 𝑦 is not free in 𝜑. The implication "to the right" is sb1 2473 and does not require the nonfreeness hypothesis. Theorem sb5 2263 replaces the nonfreeness hypothesis with a disjoint variable condition on 𝑥, 𝑦 and requires fewer axioms. Usage of this theorem is discouraged because it depends on ax-13 2367. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb6f.1 | ⊢ Ⅎ𝑦𝜑 |
Ref | Expression |
---|---|
sb5f | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6f.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | sb6f 2492 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
3 | 1 | equs45f 2454 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
4 | 2, 3 | bitr4i 278 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1532 ∃wex 1774 Ⅎwnf 1778 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-10 2130 ax-12 2167 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-ex 1775 df-nf 1779 df-sb 2061 |
This theorem is referenced by: sb7f 2520 |
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