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Theorem elsb4 2136
 Description: Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
elsb4 ([𝑦 / 𝑥]𝑧𝑥𝑧𝑦)
Distinct variable group:   𝑥,𝑧

Proof of Theorem elsb4
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-17 1506 . . . . 5 (𝑧𝑥 → ∀𝑤 𝑧𝑥)
2 elequ2 2133 . . . . 5 (𝑤 = 𝑥 → (𝑧𝑤𝑧𝑥))
31, 2sbieh 1770 . . . 4 ([𝑥 / 𝑤]𝑧𝑤𝑧𝑥)
43sbbii 1745 . . 3 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑧𝑤 ↔ [𝑦 / 𝑥]𝑧𝑥)
5 ax-17 1506 . . . 4 (𝑧𝑤 → ∀𝑥 𝑧𝑤)
65sbco2h 1944 . . 3 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑧𝑤 ↔ [𝑦 / 𝑤]𝑧𝑤)
74, 6bitr3i 185 . 2 ([𝑦 / 𝑥]𝑧𝑥 ↔ [𝑦 / 𝑤]𝑧𝑤)
8 equsb1 1765 . . . 4 [𝑦 / 𝑤]𝑤 = 𝑦
9 elequ2 2133 . . . . 5 (𝑤 = 𝑦 → (𝑧𝑤𝑧𝑦))
109sbimi 1744 . . . 4 ([𝑦 / 𝑤]𝑤 = 𝑦 → [𝑦 / 𝑤](𝑧𝑤𝑧𝑦))
118, 10ax-mp 5 . . 3 [𝑦 / 𝑤](𝑧𝑤𝑧𝑦)
12 sbbi 1939 . . 3 ([𝑦 / 𝑤](𝑧𝑤𝑧𝑦) ↔ ([𝑦 / 𝑤]𝑧𝑤 ↔ [𝑦 / 𝑤]𝑧𝑦))
1311, 12mpbi 144 . 2 ([𝑦 / 𝑤]𝑧𝑤 ↔ [𝑦 / 𝑤]𝑧𝑦)
14 ax-17 1506 . . 3 (𝑧𝑦 → ∀𝑤 𝑧𝑦)
1514sbh 1756 . 2 ([𝑦 / 𝑤]𝑧𝑦𝑧𝑦)
167, 13, 153bitri 205 1 ([𝑦 / 𝑥]𝑧𝑥𝑧𝑦)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104  [wsb 1742 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743 This theorem is referenced by: (None)
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