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Theorem sstrALT2VD 44176
Description: Virtual deduction proof of sstrALT2 44177. (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 3963 . . 3 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
2 idn1 43916 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ▶   (𝐴𝐵𝐵𝐶)   )
3 simpr 484 . . . . . . 7 ((𝐴𝐵𝐵𝐶) → 𝐵𝐶)
42, 3e1a 43969 . . . . . 6 (   (𝐴𝐵𝐵𝐶)   ▶   𝐵𝐶   )
5 simpl 482 . . . . . . . 8 ((𝐴𝐵𝐵𝐶) → 𝐴𝐵)
62, 5e1a 43969 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ▶   𝐴𝐵   )
7 idn2 43955 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐴   )
8 ssel2 3972 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
96, 7, 8e12an 44067 . . . . . 6 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐵   )
10 ssel2 3972 . . . . . 6 ((𝐵𝐶𝑥𝐵) → 𝑥𝐶)
114, 9, 10e12an 44067 . . . . 5 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐶   )
1211in2 43947 . . . 4 (   (𝐴𝐵𝐵𝐶)   ▶   (𝑥𝐴𝑥𝐶)   )
1312gen11 43958 . . 3 (   (𝐴𝐵𝐵𝐶)   ▶   𝑥(𝑥𝐴𝑥𝐶)   )
14 biimpr 219 . . 3 ((𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶)) → (∀𝑥(𝑥𝐴𝑥𝐶) → 𝐴𝐶))
151, 13, 14e01 44033 . 2 (   (𝐴𝐵𝐵𝐶)   ▶   𝐴𝐶   )
1615in1 43913 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531  wcel 2098  wss 3943
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-in 3950  df-ss 3960  df-vd1 43912  df-vd2 43920
This theorem is referenced by: (None)
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