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Theorem 2sb6rf 2451
Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) Remove variable constraints. (Revised by Wolf Lammen, 28-Sep-2018.)
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 sbequ12r 2109 . . . . 5 (𝑧 = 𝑥 → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦]𝜑))
2 sbequ12r 2109 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑦]𝜑𝜑))
31, 2sylan9bb 735 . . . 4 ((𝑧 = 𝑥𝑤 = 𝑦) → ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑𝜑))
43pm5.74i 260 . . 3 (((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
542albii 1745 . 2 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
6 2sb5rf.2 . . . . 5 𝑤𝜑
7619.23 2078 . . . 4 (∀𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
87albii 1744 . . 3 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ ∀𝑧(∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
9 2sb5rf.1 . . . 4 𝑧𝜑
10919.23 2078 . . 3 (∀𝑧(∃𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
118, 10bitri 264 . 2 (∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑))
12 2ax6e 2449 . . 3 𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦)
13 pm5.5 351 . . 3 (∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → ((∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ 𝜑))
1412, 13ax-mp 5 . 2 ((∃𝑧𝑤(𝑧 = 𝑥𝑤 = 𝑦) → 𝜑) ↔ 𝜑)
155, 11, 143bitrri 287 1 (𝜑 ↔ ∀𝑧𝑤((𝑧 = 𝑥𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wex 1701  wnf 1705  [wsb 1877
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 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
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 1878
This theorem is referenced by: (None)
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