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Theorem sstrALT2VD 40587
Description: Virtual deduction proof of sstrALT2 40588. (Contributed by Alan Sare, 11-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sstrALT2VD ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Proof of Theorem sstrALT2VD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3840 . . 3 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
2 idn1 40327 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ▶   (𝐴𝐵𝐵𝐶)   )
3 simpr 477 . . . . . . 7 ((𝐴𝐵𝐵𝐶) → 𝐵𝐶)
42, 3e1a 40380 . . . . . 6 (   (𝐴𝐵𝐵𝐶)   ▶   𝐵𝐶   )
5 simpl 475 . . . . . . . 8 ((𝐴𝐵𝐵𝐶) → 𝐴𝐵)
62, 5e1a 40380 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ▶   𝐴𝐵   )
7 idn2 40366 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐴   )
8 ssel2 3847 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
96, 7, 8e12an 40478 . . . . . 6 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐵   )
10 ssel2 3847 . . . . . 6 ((𝐵𝐶𝑥𝐵) → 𝑥𝐶)
114, 9, 10e12an 40478 . . . . 5 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐶   )
1211in2 40358 . . . 4 (   (𝐴𝐵𝐵𝐶)   ▶   (𝑥𝐴𝑥𝐶)   )
1312gen11 40369 . . 3 (   (𝐴𝐵𝐵𝐶)   ▶   𝑥(𝑥𝐴𝑥𝐶)   )
14 biimpr 212 . . 3 ((𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶)) → (∀𝑥(𝑥𝐴𝑥𝐶) → 𝐴𝐶))
151, 13, 14e01 40444 . 2 (   (𝐴𝐵𝐵𝐶)   ▶   𝐴𝐶   )
1615in1 40324 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  wal 1505  wcel 2050  wss 3823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-in 3830  df-ss 3837  df-vd1 40323  df-vd2 40331
This theorem is referenced by: (None)
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