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Mirrors > Home > ILE Home > Th. List > sbexyz | GIF version |
Description: Move existential quantifier in and out of substitution. Identical to sbex 2016 except that it has an additional disjoint variable condition on 𝑦, 𝑧. (Contributed by Jim Kingdon, 29-Dec-2017.) |
Ref | Expression |
---|---|
sbexyz | ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb5 1899 | . . 3 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑦(𝑦 = 𝑧 ∧ ∃𝑥𝜑)) | |
2 | exdistr 1921 | . . 3 ⊢ (∃𝑦∃𝑥(𝑦 = 𝑧 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝑧 ∧ ∃𝑥𝜑)) | |
3 | excom 1675 | . . 3 ⊢ (∃𝑦∃𝑥(𝑦 = 𝑧 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑦 = 𝑧 ∧ 𝜑)) | |
4 | 1, 2, 3 | 3bitr2i 208 | . 2 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥∃𝑦(𝑦 = 𝑧 ∧ 𝜑)) |
5 | sb5 1899 | . . 3 ⊢ ([𝑧 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝑧 ∧ 𝜑)) | |
6 | 5 | exbii 1616 | . 2 ⊢ (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦(𝑦 = 𝑧 ∧ 𝜑)) |
7 | 4, 6 | bitr4i 187 | 1 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∃wex 1503 [wsb 1773 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 |
This theorem depends on definitions: df-bi 117 df-sb 1774 |
This theorem is referenced by: sbex 2016 |
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