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Mirrors > Home > MPE Home > Th. List > sb56OLD | Structured version Visualization version GIF version |
Description: Obsolete version of sbalex 2233 as of 21-Sep-2024. Two equivalent ways of expressing the proper substitution of 𝑦 for 𝑥 in 𝜑, when 𝑥 and 𝑦 are distinct, namely, alternate definitions sb5 2265 and sb6 2086. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 2066. The implication "to the left" is equs4 2413 and does not require any disjoint variable condition (but the version with a disjoint variable condition, equs4v 2001, requires fewer axioms). Theorem equs45f 2456 replaces the disjoint variable condition with a nonfreeness hypothesis and equs5 2457 replaces it with a distinctor as antecedent. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2257 in place of equsex 2415 in order to remove dependency on ax-13 2369. (Revised by BJ, 20-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sb56OLD | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb5 2265 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
2 | sb6 2086 | . 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
3 | 1, 2 | bitr3i 276 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1537 ∃wex 1779 [wsb 2065 |
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 1911 ax-6 1969 ax-7 2009 ax-10 2135 ax-12 2169 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-ex 1780 df-nf 1784 df-sb 2066 |
This theorem is referenced by: (None) |
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