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Theorem 2sb5rf 1865
 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 1732 . 2 (𝜑 ↔ ∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑥]𝜑))
3 19.42v 1786 . . . 4 (∃𝑤(𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)) ↔ (𝑧 = 𝑥 ∧ ∃𝑤(𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
4 sbcom2 1863 . . . . . . 7 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
54anbi2i 430 . . . . . 6 (((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
6 anass 381 . . . . . 6 (((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑) ↔ (𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
75, 6bitri 173 . . . . 5 (((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (𝑧 = 𝑥 ∧ (𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
87exbii 1496 . . . 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 1817 . . . . . 6 ([𝑧 / 𝑥]𝜑 → ∀𝑤[𝑧 / 𝑥]𝜑)
1110sb5rf 1732 . . . . 5 ([𝑧 / 𝑥]𝜑 ↔ ∃𝑤(𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
1211anbi2i 430 . . . 4 ((𝑧 = 𝑥 ∧ [𝑧 / 𝑥]𝜑) ↔ (𝑧 = 𝑥 ∧ ∃𝑤(𝑤 = 𝑦 ∧ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
133, 8, 123bitr4ri 202 . . 3 ((𝑧 = 𝑥 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
1413exbii 1496 . 2 (∃𝑧(𝑧 = 𝑥 ∧ [𝑧 / 𝑥]𝜑) ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
152, 14bitri 173 1 (𝜑 ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  ∃wex 1381  [wsb 1645 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646 This theorem is referenced by: (None)
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