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Theorem rspsbc2VD 38570
Description: Virtual deduction proof of rspsbc2 38223. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.
1:: (   𝐴𝐵   ▶   𝐴𝐵   )
2:: (   𝐴𝐵   ,   𝐶𝐷   ▶   𝐶𝐷   )
3:: (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑥𝐵𝑦𝐷𝜑   )
4:1,3,?: e13 38454 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   [𝐴 / 𝑥]𝑦𝐷𝜑   )
5:1,4,?: e13 38454 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   𝑦𝐷[𝐴 / 𝑥]𝜑   )
6:2,5,?: e23 38461 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵 𝑦𝐷𝜑   ▶   [𝐶 / 𝑦][𝐴 / 𝑥]𝜑   )
7:6: (   𝐴𝐵   ,   𝐶𝐷   ▶   (∀𝑥𝐵 𝑦𝐷𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)   )
8:7: (   𝐴𝐵   ▶   (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑))   )
qed:8: (𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
rspsbc2VD (𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝐷,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem rspsbc2VD
StepHypRef Expression
1 idn2 38317 . . . . 5 (   𝐴𝐵   ,   𝐶𝐷   ▶   𝐶𝐷   )
2 idn1 38269 . . . . . 6 (   𝐴𝐵   ▶   𝐴𝐵   )
3 idn3 38319 . . . . . . 7 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   𝑥𝐵𝑦𝐷 𝜑   )
4 rspsbc 3499 . . . . . . 7 (𝐴𝐵 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐴 / 𝑥]𝑦𝐷 𝜑))
52, 3, 4e13 38454 . . . . . 6 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   [𝐴 / 𝑥]𝑦𝐷 𝜑   )
6 sbcralg 3495 . . . . . . 7 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐷 𝜑 ↔ ∀𝑦𝐷 [𝐴 / 𝑥]𝜑))
76biimpd 219 . . . . . 6 (𝐴𝐵 → ([𝐴 / 𝑥]𝑦𝐷 𝜑 → ∀𝑦𝐷 [𝐴 / 𝑥]𝜑))
82, 5, 7e13 38454 . . . . 5 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   𝑦𝐷 [𝐴 / 𝑥]𝜑   )
9 rspsbc 3499 . . . . 5 (𝐶𝐷 → (∀𝑦𝐷 [𝐴 / 𝑥]𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑))
101, 8, 9e23 38461 . . . 4 (   𝐴𝐵   ,   𝐶𝐷   ,   𝑥𝐵𝑦𝐷 𝜑   ▶   [𝐶 / 𝑦][𝐴 / 𝑥]𝜑   )
1110in3 38313 . . 3 (   𝐴𝐵   ,   𝐶𝐷   ▶   (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)   )
1211in2 38309 . 2 (   𝐴𝐵   ▶   (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑))   )
1312in1 38266 1 (𝐴𝐵 → (𝐶𝐷 → (∀𝑥𝐵𝑦𝐷 𝜑[𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  wral 2907  [wsbc 3417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-sbc 3418  df-vd1 38265  df-vd2 38273  df-vd3 38285
This theorem is referenced by: (None)
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