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Theorem 2sb5rf 1879
Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.)
Hypotheses
Ref Expression
2sb5rf.1 (𝜑 → ∀𝑧𝜑)
2sb5rf.2 (𝜑 → ∀𝑤𝜑)
Assertion
Ref Expression
2sb5rf (𝜑 ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Distinct variable groups:   𝑥,𝑦   𝑥,𝑤   𝑦,𝑧   𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 2sb5rf
StepHypRef Expression
1 2sb5rf.1 . . 3 (𝜑 → ∀𝑧𝜑)
21sb5rf 1746 . 2 (𝜑 ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑥]𝜑))
3 19.42v 1800 . . . 4 (∃𝑤(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)) ↔ (𝑧 = 𝑥 ∧ ∃𝑤(𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
4 sbcom2 1877 . . . . . . 7 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
54anbi2i 438 . . . . . 6 (((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
6 anass 387 . . . . . 6 (((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑) ↔ (𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
75, 6bitri 177 . . . . 5 (((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
87exbii 1510 . . . 4 (∃𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ∃𝑤(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
9 2sb5rf.2 . . . . . . 7 (𝜑 → ∀𝑤𝜑)
109hbsbv 1831 . . . . . 6 ([𝑧 / 𝑥]𝜑 → ∀𝑤[𝑧 / 𝑥]𝜑)
1110sb5rf 1746 . . . . 5 ([𝑧 / 𝑥]𝜑 ↔ ∃𝑤(𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
1211anbi2i 438 . . . 4 ((𝑧 = 𝑥 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧 = 𝑥 ∧ ∃𝑤(𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
133, 8, 123bitr4ri 206 . . 3 ((𝑧 = 𝑥 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
1413exbii 1510 . 2 (∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
152, 14bitri 177 1 (𝜑 ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1255  wex 1395  [wsb 1659
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-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442
This theorem depends on definitions:  df-bi 114  df-nf 1364  df-sb 1660
This theorem is referenced by: (None)
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