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Theorem sbexyz 1895
Description: Move existential quantifier in and out of substitution. Identical to sbex 1896 except that it has an additional distinct variable constraint on 𝑦 and 𝑧. (Contributed by Jim Kingdon, 29-Dec-2017.)
Assertion
Ref Expression
sbexyz ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbexyz
StepHypRef Expression
1 sb5 1783 . . 3 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑦(𝑦 = 𝑧 ∧ ∃𝑥𝜑))
2 exdistr 1803 . . 3 (∃𝑦𝑥(𝑦 = 𝑧𝜑) ↔ ∃𝑦(𝑦 = 𝑧 ∧ ∃𝑥𝜑))
3 excom 1570 . . 3 (∃𝑦𝑥(𝑦 = 𝑧𝜑) ↔ ∃𝑥𝑦(𝑦 = 𝑧𝜑))
41, 2, 33bitr2i 201 . 2 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥𝑦(𝑦 = 𝑧𝜑))
5 sb5 1783 . . 3 ([𝑧 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝑧𝜑))
65exbii 1512 . 2 (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥𝑦(𝑦 = 𝑧𝜑))
74, 6bitr4i 180 1 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wex 1397  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  sbex  1896
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