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Theorem 2stdpc5 32494
Description: A double stdpc5 2074 (one direction of PM*11.3). See also 2stdpc4 2353 and 19.21vv 38092. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
2stdpc5.1 𝑥𝜑
2stdpc5.2 𝑦𝜑
Assertion
Ref Expression
2stdpc5 (∀𝑥𝑦(𝜑𝜓) → (𝜑 → ∀𝑥𝑦𝜓))

Proof of Theorem 2stdpc5
StepHypRef Expression
1 2stdpc5.2 . . . 4 𝑦𝜑
21stdpc5 2074 . . 3 (∀𝑦(𝜑𝜓) → (𝜑 → ∀𝑦𝜓))
32alimi 1736 . 2 (∀𝑥𝑦(𝜑𝜓) → ∀𝑥(𝜑 → ∀𝑦𝜓))
4 2stdpc5.1 . . 3 𝑥𝜑
54stdpc5 2074 . 2 (∀𝑥(𝜑 → ∀𝑦𝜓) → (𝜑 → ∀𝑥𝑦𝜓))
63, 5syl 17 1 (∀𝑥𝑦(𝜑𝜓) → (𝜑 → ∀𝑥𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wnf 1705
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-12 2044
This theorem depends on definitions:  df-bi 197  df-ex 1702  df-nf 1707
This theorem is referenced by:  ax11-pm  32497  ax11-pm2  32501
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