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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 1877 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 1764 | . . 3 ⊢ ([z / y]∃xφ ↔ ∃y(y = z ∧ ∃xφ)) | |
2 | exdistr 1784 | . . 3 ⊢ (∃y∃x(y = z ∧ φ) ↔ ∃y(y = z ∧ ∃xφ)) | |
3 | excom 1551 | . . 3 ⊢ (∃y∃x(y = z ∧ φ) ↔ ∃x∃y(y = z ∧ φ)) | |
4 | 1, 2, 3 | 3bitr2i 197 | . 2 ⊢ ([z / y]∃xφ ↔ ∃x∃y(y = z ∧ φ)) |
5 | sb5 1764 | . . 3 ⊢ ([z / y]φ ↔ ∃y(y = z ∧ φ)) | |
6 | 5 | exbii 1493 | . 2 ⊢ (∃x[z / y]φ ↔ ∃x∃y(y = z ∧ φ)) |
7 | 4, 6 | bitr4i 176 | 1 ⊢ ([z / y]∃xφ ↔ ∃x[z / y]φ) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∃wex 1378 [wsb 1642 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 |
This theorem depends on definitions: df-bi 110 df-sb 1643 |
This theorem is referenced by: sbex 1877 |
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