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Theorem sstrALT2VD 44854
Description: Virtual deduction proof of sstrALT2 44855. (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 df-ss 3968 . . 3 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
2 idn1 44594 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ▶   (𝐴𝐵𝐵𝐶)   )
3 simpr 484 . . . . . . 7 ((𝐴𝐵𝐵𝐶) → 𝐵𝐶)
42, 3e1a 44647 . . . . . 6 (   (𝐴𝐵𝐵𝐶)   ▶   𝐵𝐶   )
5 simpl 482 . . . . . . . 8 ((𝐴𝐵𝐵𝐶) → 𝐴𝐵)
62, 5e1a 44647 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ▶   𝐴𝐵   )
7 idn2 44633 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐴   )
8 ssel2 3978 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
96, 7, 8e12an 44745 . . . . . 6 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐵   )
10 ssel2 3978 . . . . . 6 ((𝐵𝐶𝑥𝐵) → 𝑥𝐶)
114, 9, 10e12an 44745 . . . . 5 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐶   )
1211in2 44625 . . . 4 (   (𝐴𝐵𝐵𝐶)   ▶   (𝑥𝐴𝑥𝐶)   )
1312gen11 44636 . . 3 (   (𝐴𝐵𝐵𝐶)   ▶   𝑥(𝑥𝐴𝑥𝐶)   )
14 biimpr 220 . . 3 ((𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶)) → (∀𝑥(𝑥𝐴𝑥𝐶) → 𝐴𝐶))
151, 13, 14e01 44711 . 2 (   (𝐴𝐵𝐵𝐶)   ▶   𝐴𝐶   )
1615in1 44591 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  wcel 2108  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-clel 2816  df-ss 3968  df-vd1 44590  df-vd2 44598
This theorem is referenced by: (None)
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