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Theorem sbex 1935
Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)
Assertion
Ref Expression
sbex ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
Distinct variable groups:   𝑥,𝑦   𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbex
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 sbexyz 1934 . . . 4 ([𝑤 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑤 / 𝑦]𝜑)
21sbbii 1702 . . 3 ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑤]∃𝑥[𝑤 / 𝑦]𝜑)
3 sbexyz 1934 . . 3 ([𝑧 / 𝑤]∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑)
42, 3bitri 183 . 2 ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑)
5 ax-17 1471 . . 3 (∃𝑥𝜑 → ∀𝑤𝑥𝜑)
65sbco2v 1876 . 2 ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑦]∃𝑥𝜑)
7 ax-17 1471 . . . 4 (𝜑 → ∀𝑤𝜑)
87sbco2v 1876 . . 3 ([𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑)
98exbii 1548 . 2 (∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
104, 6, 93bitr3i 209 1 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
Colors of variables: wff set class
Syntax hints:  wb 104  wex 1433  [wsb 1699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480
This theorem depends on definitions:  df-bi 116  df-nf 1402  df-sb 1700
This theorem is referenced by:  sbabel  2261  sbcex2  2906  sbcexg  2907
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