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Theorem 2sb5rf 2453
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.1 . . . 4 𝑧𝜑
2119.41 2202 . . 3 (∃𝑧(∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑) ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
3 2sb5rf.2 . . . . 5 𝑤𝜑
4319.41 2202 . . . 4 (∃𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑) ↔ (∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
54exbii 1829 . . 3 (∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑) ↔ ∃𝑧(∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
6 2ax6e 2451 . . . 4 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
76biantrur 531 . . 3 (𝜑 ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
82, 5, 73bitr4ri 305 . 2 (𝜑 ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
9 sbequ12r 2217 . . . . 5 (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑))
10 sbequ12r 2217 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜑𝜑))
119, 10sylan9bb 510 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑𝜑))
1211pm5.32i 575 . . 3 (((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
13122exbii 1830 . 2 (∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ 𝜑))
148, 13bitr4i 279 1 (𝜑 ↔ ∃𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wex 1761  wnf 1765  [wsb 2042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043
This theorem is referenced by:  sbel2x  2456
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