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Mirrors > Home > MPE Home > Th. List > 2sb5rf | Structured version Visualization version GIF version |
Description: Reversed double substitution. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove distinct variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2sb5rf.1 | ⊢ Ⅎ𝑧𝜑 |
2sb5rf.2 | ⊢ Ⅎ𝑤𝜑 |
Ref | Expression |
---|---|
2sb5rf | ⊢ (𝜑 ↔ ∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2sb5rf.1 | . . . 4 ⊢ Ⅎ𝑧𝜑 | |
2 | 1 | 19.41 2233 | . . 3 ⊢ (∃𝑧(∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ (∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑)) |
3 | 2sb5rf.2 | . . . . 5 ⊢ Ⅎ𝑤𝜑 | |
4 | 3 | 19.41 2233 | . . . 4 ⊢ (∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ (∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑)) |
5 | 4 | exbii 1855 | . . 3 ⊢ (∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑧(∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑)) |
6 | 2ax6e 2470 | . . . 4 ⊢ ∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) | |
7 | 6 | biantrur 534 | . . 3 ⊢ (𝜑 ↔ (∃𝑧∃𝑤(𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑)) |
8 | 2, 5, 7 | 3bitr4ri 307 | . 2 ⊢ (𝜑 ↔ ∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑)) |
9 | sbequ12r 2250 | . . . . 5 ⊢ (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑)) | |
10 | sbequ12r 2250 | . . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜑 ↔ 𝜑)) | |
11 | 9, 10 | sylan9bb 513 | . . . 4 ⊢ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ 𝜑)) |
12 | 11 | pm5.32i 578 | . . 3 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑)) |
13 | 12 | 2exbii 1856 | . 2 ⊢ (∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ 𝜑)) |
14 | 8, 13 | bitr4i 281 | 1 ⊢ (𝜑 ↔ ∃𝑧∃𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∃wex 1787 Ⅎwnf 1791 [wsb 2070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-11 2158 ax-12 2175 ax-13 2371 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 |
This theorem is referenced by: sbel2x 2473 |
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