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Mirrors > Home > MPE Home > Th. List > 2sb5 | Structured version Visualization version GIF version |
Description: Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.) |
Ref | Expression |
---|---|
2sb5 | ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb5 2267 | . 2 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝑧 ∧ [𝑤 / 𝑦]𝜑)) | |
2 | 19.42v 1945 | . . . 4 ⊢ (∃𝑦(𝑥 = 𝑧 ∧ (𝑦 = 𝑤 ∧ 𝜑)) ↔ (𝑥 = 𝑧 ∧ ∃𝑦(𝑦 = 𝑤 ∧ 𝜑))) | |
3 | anass 469 | . . . . 5 ⊢ (((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑) ↔ (𝑥 = 𝑧 ∧ (𝑦 = 𝑤 ∧ 𝜑))) | |
4 | 3 | exbii 1839 | . . . 4 ⊢ (∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝑧 ∧ (𝑦 = 𝑤 ∧ 𝜑))) |
5 | sb5 2267 | . . . . 5 ⊢ ([𝑤 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝑤 ∧ 𝜑)) | |
6 | 5 | anbi2i 622 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ [𝑤 / 𝑦]𝜑) ↔ (𝑥 = 𝑧 ∧ ∃𝑦(𝑦 = 𝑤 ∧ 𝜑))) |
7 | 2, 4, 6 | 3bitr4ri 305 | . . 3 ⊢ ((𝑥 = 𝑧 ∧ [𝑤 / 𝑦]𝜑) ↔ ∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) |
8 | 7 | exbii 1839 | . 2 ⊢ (∃𝑥(𝑥 = 𝑧 ∧ [𝑤 / 𝑦]𝜑) ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) |
9 | 1, 8 | bitri 276 | 1 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∃wex 1771 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 ax-12 2167 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 |
This theorem is referenced by: opelopabsbALT 5407 |
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