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Theorem 2sb6rf 2487
 Description: Reversed double substitution. Usage of this theorem is discouraged because it depends on ax-13 2380. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.) (Proof shortened by Wolf Lammen, 13-Apr-2023.) (New usage is discouraged.)
Hypotheses
Ref Expression
2sb5rf.1 𝑧𝜑
2sb5rf.2 𝑤𝜑
Assertion
Ref Expression
2sb6rf (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 2sb6rf
StepHypRef Expression
1 2sb5rf.1 . . . 4 𝑧𝜑
2119.23 2210 . . 3 (∀𝑧(∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
3 2sb5rf.2 . . . . 5 𝑤𝜑
4319.23 2210 . . . 4 (∀𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
54albii 1822 . . 3 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ ∀𝑧(∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
6 2ax6e 2484 . . . 4 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
76a1bi 367 . . 3 (𝜑 ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
82, 5, 73bitr4ri 308 . 2 (𝜑 ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
9 sbequ12r 2252 . . . . 5 (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑))
10 sbequ12r 2252 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜑𝜑))
119, 10sylan9bb 514 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑𝜑))
1211pm5.74i 274 . . 3 (((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
13122albii 1823 . 2 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
148, 13bitr4i 281 1 (𝜑 ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1537  ∃wex 1782  Ⅎwnf 1786  [wsb 2070 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2380 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071 This theorem is referenced by: (None)
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