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Theorem sstrALT2VD 39568
Description: Virtual deduction proof of sstrALT2 39569. (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 3732 . . 3 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
2 idn1 39292 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ▶   (𝐴𝐵𝐵𝐶)   )
3 simpr 479 . . . . . . 7 ((𝐴𝐵𝐵𝐶) → 𝐵𝐶)
42, 3e1a 39354 . . . . . 6 (   (𝐴𝐵𝐵𝐶)   ▶   𝐵𝐶   )
5 simpl 474 . . . . . . . 8 ((𝐴𝐵𝐵𝐶) → 𝐴𝐵)
62, 5e1a 39354 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ▶   𝐴𝐵   )
7 idn2 39340 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐴   )
8 ssel2 3739 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
96, 7, 8e12an 39454 . . . . . 6 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐵   )
10 ssel2 3739 . . . . . 6 ((𝐵𝐶𝑥𝐵) → 𝑥𝐶)
114, 9, 10e12an 39454 . . . . 5 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐶   )
1211in2 39332 . . . 4 (   (𝐴𝐵𝐵𝐶)   ▶   (𝑥𝐴𝑥𝐶)   )
1312gen11 39343 . . 3 (   (𝐴𝐵𝐵𝐶)   ▶   𝑥(𝑥𝐴𝑥𝐶)   )
14 biimpr 210 . . 3 ((𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶)) → (∀𝑥(𝑥𝐴𝑥𝐶) → 𝐴𝐶))
151, 13, 14e01 39418 . 2 (   (𝐴𝐵𝐵𝐶)   ▶   𝐴𝐶   )
1615in1 39289 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630  wcel 2139  wss 3715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-in 3722  df-ss 3729  df-vd1 39288  df-vd2 39296
This theorem is referenced by: (None)
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