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Theorem elex22VD 41034
Description: Virtual deduction proof of elex22 3522. (Contributed by Alan Sare, 24-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
elex22VD ((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem elex22VD
StepHypRef Expression
1 idn1 40769 . . . . 5 (   (𝐴𝐵𝐴𝐶)   ▶   (𝐴𝐵𝐴𝐶)   )
2 simpl 483 . . . . 5 ((𝐴𝐵𝐴𝐶) → 𝐴𝐵)
31, 2e1a 40822 . . . 4 (   (𝐴𝐵𝐴𝐶)   ▶   𝐴𝐵   )
4 elisset 3510 . . . 4 (𝐴𝐵 → ∃𝑥 𝑥 = 𝐴)
53, 4e1a 40822 . . 3 (   (𝐴𝐵𝐴𝐶)   ▶   𝑥 𝑥 = 𝐴   )
6 idn2 40808 . . . . . . . 8 (   (𝐴𝐵𝐴𝐶)   ,   𝑥 = 𝐴   ▶   𝑥 = 𝐴   )
7 eleq1a 2912 . . . . . . . 8 (𝐴𝐵 → (𝑥 = 𝐴𝑥𝐵))
83, 6, 7e12 40919 . . . . . . 7 (   (𝐴𝐵𝐴𝐶)   ,   𝑥 = 𝐴   ▶   𝑥𝐵   )
9 simpr 485 . . . . . . . . 9 ((𝐴𝐵𝐴𝐶) → 𝐴𝐶)
101, 9e1a 40822 . . . . . . . 8 (   (𝐴𝐵𝐴𝐶)   ▶   𝐴𝐶   )
11 eleq1a 2912 . . . . . . . 8 (𝐴𝐶 → (𝑥 = 𝐴𝑥𝐶))
1210, 6, 11e12 40919 . . . . . . 7 (   (𝐴𝐵𝐴𝐶)   ,   𝑥 = 𝐴   ▶   𝑥𝐶   )
13 pm3.2 470 . . . . . . 7 (𝑥𝐵 → (𝑥𝐶 → (𝑥𝐵𝑥𝐶)))
148, 12, 13e22 40866 . . . . . 6 (   (𝐴𝐵𝐴𝐶)   ,   𝑥 = 𝐴   ▶   (𝑥𝐵𝑥𝐶)   )
1514in2 40800 . . . . 5 (   (𝐴𝐵𝐴𝐶)   ▶   (𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶))   )
1615gen11 40811 . . . 4 (   (𝐴𝐵𝐴𝐶)   ▶   𝑥(𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶))   )
17 exim 1827 . . . 4 (∀𝑥(𝑥 = 𝐴 → (𝑥𝐵𝑥𝐶)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥𝐵𝑥𝐶)))
1816, 17e1a 40822 . . 3 (   (𝐴𝐵𝐴𝐶)   ▶   (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥𝐵𝑥𝐶))   )
19 pm2.27 42 . . 3 (∃𝑥 𝑥 = 𝐴 → ((∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥𝐵𝑥𝐶)) → ∃𝑥(𝑥𝐵𝑥𝐶)))
205, 18, 19e11 40883 . 2 (   (𝐴𝐵𝐴𝐶)   ▶   𝑥(𝑥𝐵𝑥𝐶)   )
2120in1 40766 1 ((𝐴𝐵𝐴𝐶) → ∃𝑥(𝑥𝐵𝑥𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1528   = wceq 1530  wex 1773  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-cleq 2818  df-clel 2897  df-vd1 40765  df-vd2 40773
This theorem is referenced by: (None)
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