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Theorem sbex 1880
 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 1879 . . . 4 ([𝑤 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑤 / 𝑦]𝜑)
21sbbii 1648 . . 3 ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑤]∃𝑥[𝑤 / 𝑦]𝜑)
3 sbexyz 1879 . . 3 ([𝑧 / 𝑤]∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑)
42, 3bitri 173 . 2 ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑)
5 ax-17 1419 . . 3 (∃𝑥𝜑 → ∀𝑤𝑥𝜑)
65sbco2v 1821 . 2 ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑦]∃𝑥𝜑)
7 ax-17 1419 . . . 4 (𝜑 → ∀𝑤𝜑)
87sbco2v 1821 . . 3 ([𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑)
98exbii 1496 . 2 (∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
104, 6, 93bitr3i 199 1 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∃wex 1381  [wsb 1645 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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by:  sbabel  2203  sbcex2  2812  sbcexg  2813
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