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Theorem 2stdpc4 2353
Description: A double specialization using explicit substitution. This is Theorem PM*11.1 in [WhiteheadRussell] p. 159. See stdpc4 2352 for the analogous single specialization. See 2sp 2055 for another double specialization. (Contributed by Andrew Salmon, 24-May-2011.) (Revised by BJ, 21-Oct-2018.)
Assertion
Ref Expression
2stdpc4 (∀𝑥𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Proof of Theorem 2stdpc4
StepHypRef Expression
1 stdpc4 2352 . . 3 (∀𝑦𝜑 → [𝑤 / 𝑦]𝜑)
21alimi 1738 . 2 (∀𝑥𝑦𝜑 → ∀𝑥[𝑤 / 𝑦]𝜑)
3 stdpc4 2352 . 2 (∀𝑥[𝑤 / 𝑦]𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
42, 3syl 17 1 (∀𝑥𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1480  [wsb 1879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-12 2046  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1704  df-sb 1880
This theorem is referenced by:  ax11-pm2  32807  pm11.11  38399
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