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Mirrors > Home > MPE Home > Th. List > sbel2x | Structured version Visualization version GIF version |
Description: Elimination of double substitution. Usage of this theorem is discouraged because it depends on ax-13 2363. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 29-Sep-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sbel2x | ⊢ (𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1909 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1909 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
3 | 1, 2 | 2sb5rf 2463 | . 2 ⊢ (𝜑 ↔ ∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑)) |
4 | ancom 460 | . . . 4 ⊢ ((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ↔ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) | |
5 | 4 | anbi1i 623 | . . 3 ⊢ (((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑) ↔ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑)) |
6 | 5 | 2exbii 1843 | . 2 ⊢ (∃𝑦∃𝑥((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑) ↔ ∃𝑦∃𝑥((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑)) |
7 | excom 2154 | . 2 ⊢ (∃𝑦∃𝑥((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑) ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑)) | |
8 | 3, 6, 7 | 3bitri 297 | 1 ⊢ (𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) ∧ [𝑦 / 𝑤][𝑥 / 𝑧]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∃wex 1773 [wsb 2059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-10 2129 ax-11 2146 ax-12 2163 ax-13 2363 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 |
This theorem is referenced by: (None) |
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