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Theorem 2sb5rf 2455
Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove distinct variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.)
Hypotheses
Ref Expression
2sb5rf.1 𝑧𝜑
2sb5rf.2 𝑤𝜑
Assertion
Ref Expression
2sb5rf (𝜑 ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Distinct variable group:   𝑧,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 2sb5rf
StepHypRef Expression
1 2sb5rf.2 . . . . 5 𝑤𝜑
2119.41 2106 . . . 4 (∃𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑) ↔ (∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
32exbii 1772 . . 3 (∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑧(∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
4 2sb5rf.1 . . . 4 𝑧𝜑
5419.41 2106 . . 3 (∃𝑧(∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑) ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
63, 5bitri 264 . 2 (∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑) ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
7 sbequ12r 2114 . . . . 5 (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑))
8 sbequ12r 2114 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜑𝜑))
97, 8sylan9bb 735 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑𝜑))
109pm5.32i 668 . . 3 (((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
11102exbii 1773 . 2 (∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
12 2ax6e 2454 . . 3 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
1312biantrur 527 . 2 (𝜑 ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
146, 11, 133bitr4ri 293 1 (𝜑 ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wex 1701  wnf 1705  [wsb 1882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883
This theorem is referenced by:  sbel2x  2463
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