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Theorem 19.9d2r 30245
Description: A deduction version of one direction of 19.9 2207 with two variables. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypotheses
Ref Expression
19.9d2r.1 (𝜑 → Ⅎ𝑥𝜓)
19.9d2r.2 (𝜑 → Ⅎ𝑦𝜓)
19.9d2r.3 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)
Assertion
Ref Expression
19.9d2r (𝜑𝜓)
Distinct variable group:   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem 19.9d2r
StepHypRef Expression
1 nfv 1916 . 2 𝑦𝜑
2 19.9d2r.1 . 2 (𝜑 → Ⅎ𝑥𝜓)
3 19.9d2r.2 . 2 (𝜑 → Ⅎ𝑦𝜓)
4 19.9d2r.3 . 2 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)
51, 2, 3, 419.9d2rf 30244 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wnf 1785  wrex 3134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-rex 3139
This theorem is referenced by:  xrofsup  30503
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