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Theorem sstrALT2VD 39566
 Description: Virtual deduction proof of sstrALT2 39567. (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 3730 . . 3 (𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶))
2 idn1 39290 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ▶   (𝐴𝐵𝐵𝐶)   )
3 simpr 479 . . . . . . 7 ((𝐴𝐵𝐵𝐶) → 𝐵𝐶)
42, 3e1a 39352 . . . . . 6 (   (𝐴𝐵𝐵𝐶)   ▶   𝐵𝐶   )
5 simpl 474 . . . . . . . 8 ((𝐴𝐵𝐵𝐶) → 𝐴𝐵)
62, 5e1a 39352 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ▶   𝐴𝐵   )
7 idn2 39338 . . . . . . 7 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐴   )
8 ssel2 3737 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
96, 7, 8e12an 39452 . . . . . 6 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐵   )
10 ssel2 3737 . . . . . 6 ((𝐵𝐶𝑥𝐵) → 𝑥𝐶)
114, 9, 10e12an 39452 . . . . 5 (   (𝐴𝐵𝐵𝐶)   ,   𝑥𝐴   ▶   𝑥𝐶   )
1211in2 39330 . . . 4 (   (𝐴𝐵𝐵𝐶)   ▶   (𝑥𝐴𝑥𝐶)   )
1312gen11 39341 . . 3 (   (𝐴𝐵𝐵𝐶)   ▶   𝑥(𝑥𝐴𝑥𝐶)   )
14 biimpr 210 . . 3 ((𝐴𝐶 ↔ ∀𝑥(𝑥𝐴𝑥𝐶)) → (∀𝑥(𝑥𝐴𝑥𝐶) → 𝐴𝐶))
151, 13, 14e01 39416 . 2 (   (𝐴𝐵𝐵𝐶)   ▶   𝐴𝐶   )
1615in1 39287 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1628   ∈ wcel 2137   ⊆ wss 3713 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-in 3720  df-ss 3727  df-vd1 39286  df-vd2 39294 This theorem is referenced by: (None)
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