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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 1921 except that it has an additional distinct variable constraint on 𝑦 and 𝑧. (Contributed by Jim Kingdon, 29-Dec-2017.) |
Ref | Expression |
---|---|
sbexyz | ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb5 1808 | . . 3 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑦(𝑦 = 𝑧 ∧ ∃𝑥𝜑)) | |
2 | exdistr 1828 | . . 3 ⊢ (∃𝑦∃𝑥(𝑦 = 𝑧 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝑧 ∧ ∃𝑥𝜑)) | |
3 | excom 1594 | . . 3 ⊢ (∃𝑦∃𝑥(𝑦 = 𝑧 ∧ 𝜑) ↔ ∃𝑥∃𝑦(𝑦 = 𝑧 ∧ 𝜑)) | |
4 | 1, 2, 3 | 3bitr2i 206 | . 2 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥∃𝑦(𝑦 = 𝑧 ∧ 𝜑)) |
5 | sb5 1808 | . . 3 ⊢ ([𝑧 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝑧 ∧ 𝜑)) | |
6 | 5 | exbii 1536 | . 2 ⊢ (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦(𝑦 = 𝑧 ∧ 𝜑)) |
7 | 4, 6 | bitr4i 185 | 1 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∃wex 1421 [wsb 1685 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 |
This theorem depends on definitions: df-bi 115 df-sb 1686 |
This theorem is referenced by: sbex 1921 |
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