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Theorem 19.9d2rf 29446
Description: A deduction version of one direction of 19.9 2110 with two variables. (Contributed by Thierry Arnoux, 20-Mar-2017.)
Hypotheses
Ref Expression
19.9d2rf.0 𝑦𝜑
19.9d2rf.1 (𝜑 → Ⅎ𝑥𝜓)
19.9d2rf.2 (𝜑 → Ⅎ𝑦𝜓)
19.9d2rf.3 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)
Assertion
Ref Expression
19.9d2rf (𝜑𝜓)

Proof of Theorem 19.9d2rf
StepHypRef Expression
1 19.9d2rf.3 . . . 4 (𝜑 → ∃𝑥𝐴𝑦𝐵 𝜓)
2 rexex 3031 . . . 4 (∃𝑥𝐴𝑦𝐵 𝜓 → ∃𝑥𝑦𝐵 𝜓)
3 rexex 3031 . . . . 5 (∃𝑦𝐵 𝜓 → ∃𝑦𝜓)
43eximi 1802 . . . 4 (∃𝑥𝑦𝐵 𝜓 → ∃𝑥𝑦𝜓)
51, 2, 43syl 18 . . 3 (𝜑 → ∃𝑥𝑦𝜓)
6 19.9d2rf.0 . . . . 5 𝑦𝜑
7 19.9d2rf.1 . . . . 5 (𝜑 → Ⅎ𝑥𝜓)
86, 7nfexd 2203 . . . 4 (𝜑 → Ⅎ𝑥𝑦𝜓)
9819.9d 2108 . . 3 (𝜑 → (∃𝑥𝑦𝜓 → ∃𝑦𝜓))
105, 9mpd 15 . 2 (𝜑 → ∃𝑦𝜓)
11 19.9d2rf.2 . . 3 (𝜑 → Ⅎ𝑦𝜓)
121119.9d 2108 . 2 (𝜑 → (∃𝑦𝜓𝜓))
1310, 12mpd 15 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1744  wnf 1748  wrex 2942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-rex 2947
This theorem is referenced by:  19.9d2r  29447  xrofsup  29661
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