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Mirrors > Home > MPE Home > Th. List > dfsb7OLD | Structured version Visualization version GIF version |
Description: Obsolete version of dfsb7 2281 as of 3-Sep-2023. (Contributed by NM, 28-Jan-2004.) Revise df-sb 2070. (Revised by BJ, 25-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dfsb7OLD | ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sb 2070 | . 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
2 | sb56 2274 | . 2 ⊢ (∃𝑦(𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑))) | |
3 | sb56 2274 | . . . . 5 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)) | |
4 | 3 | bicomi 227 | . . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
5 | 4 | anbi2i 625 | . . 3 ⊢ ((𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
6 | 5 | exbii 1849 | . 2 ⊢ (∃𝑦(𝑦 = 𝑡 ∧ ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
7 | 1, 2, 6 | 3bitr2i 302 | 1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1536 ∃wex 1781 [wsb 2069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-ex 1782 df-nf 1786 df-sb 2070 |
This theorem is referenced by: (None) |
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