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Theorem sbex 2555
Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)
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
StepHypRef Expression
1 sbn 2482 . . 3 ([𝑧 / 𝑦] ¬ ∀𝑥 ¬ 𝜑 ↔ ¬ [𝑧 / 𝑦]∀𝑥 ¬ 𝜑)
2 sbal 2554 . . . 4 ([𝑧 / 𝑦]∀𝑥 ¬ 𝜑 ↔ ∀𝑥[𝑧 / 𝑦] ¬ 𝜑)
3 sbn 2482 . . . . 5 ([𝑧 / 𝑦] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑦]𝜑)
43albii 1914 . . . 4 (∀𝑥[𝑧 / 𝑦] ¬ 𝜑 ↔ ∀𝑥 ¬ [𝑧 / 𝑦]𝜑)
52, 4bitri 266 . . 3 ([𝑧 / 𝑦]∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ [𝑧 / 𝑦]𝜑)
61, 5xchbinx 325 . 2 ([𝑧 / 𝑦] ¬ ∀𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ [𝑧 / 𝑦]𝜑)
7 df-ex 1875 . . 3 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
87sbbii 2069 . 2 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑦] ¬ ∀𝑥 ¬ 𝜑)
9 df-ex 1875 . 2 (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ¬ ∀𝑥 ¬ [𝑧 / 𝑦]𝜑)
106, 8, 93bitr4i 294 1 ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wal 1650  wex 1874  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063
This theorem is referenced by:  sbmo  2636  sbabel  2936  sbcex2  3649
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