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Theorem elsb3 1949
 Description: Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
elsb3 ([𝑦 / 𝑥]𝑥𝑧𝑦𝑧)
Distinct variable group:   𝑥,𝑧

Proof of Theorem elsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 ax-17 1506 . . . . 5 (𝑥𝑧 → ∀𝑤 𝑥𝑧)
2 elequ1 1690 . . . . 5 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
31, 2sbieh 1763 . . . 4 ([𝑥 / 𝑤]𝑤𝑧𝑥𝑧)
43sbbii 1738 . . 3 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤𝑧 ↔ [𝑦 / 𝑥]𝑥𝑧)
5 ax-17 1506 . . . 4 (𝑤𝑧 → ∀𝑥 𝑤𝑧)
65sbco2h 1935 . . 3 ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤𝑧 ↔ [𝑦 / 𝑤]𝑤𝑧)
74, 6bitr3i 185 . 2 ([𝑦 / 𝑥]𝑥𝑧 ↔ [𝑦 / 𝑤]𝑤𝑧)
8 equsb1 1758 . . . 4 [𝑦 / 𝑤]𝑤 = 𝑦
9 elequ1 1690 . . . . 5 (𝑤 = 𝑦 → (𝑤𝑧𝑦𝑧))
109sbimi 1737 . . . 4 ([𝑦 / 𝑤]𝑤 = 𝑦 → [𝑦 / 𝑤](𝑤𝑧𝑦𝑧))
118, 10ax-mp 5 . . 3 [𝑦 / 𝑤](𝑤𝑧𝑦𝑧)
12 sbbi 1930 . . 3 ([𝑦 / 𝑤](𝑤𝑧𝑦𝑧) ↔ ([𝑦 / 𝑤]𝑤𝑧 ↔ [𝑦 / 𝑤]𝑦𝑧))
1311, 12mpbi 144 . 2 ([𝑦 / 𝑤]𝑤𝑧 ↔ [𝑦 / 𝑤]𝑦𝑧)
14 ax-17 1506 . . 3 (𝑦𝑧 → ∀𝑤 𝑦𝑧)
1514sbh 1749 . 2 ([𝑦 / 𝑤]𝑦𝑧𝑦𝑧)
167, 13, 153bitri 205 1 ([𝑦 / 𝑥]𝑥𝑧𝑦𝑧)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104  [wsb 1735 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736 This theorem is referenced by:  cvjust  2132
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