Theorem rspsbc2VD 40680

Description: Virtual deduction proof of rspsbc2 40359. 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::	⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
2::	⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ▶   𝐶 ∈ 𝐷   )
3::	⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑   )
4:1,3,?: e13 40573	⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   [𝐴 / 𝑥]∀𝑦 ∈ 𝐷𝜑   )
5:1,4,?: e13 40573	⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑦 ∈ 𝐷[𝐴 / 𝑥]𝜑   )
6:2,5,?: e23 40580	⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   [𝐶 / 𝑦][𝐴 / 𝑥]𝜑   )
7:6:	⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ▶   (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)   )
8:7:	⊢ (   𝐴 ∈ 𝐵   ▶   (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))   )
qed:8:	⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵∀𝑦 ∈ 𝐷𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rspsbc2VD	⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 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

Step	Hyp	Ref	Expression
1		idn2 40438	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ▶   𝐶 ∈ 𝐷   )
2		idn1 40399	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
3		idn3 40440	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   )
4		rspsbc 3785	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑))
5	2, 3, 4	e13 40573	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   [𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑   )
6		sbcralg 3780	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑))
7	6	biimpd 230	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑))
8	2, 5, 7	e13 40573	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑   )
9		rspsbc 3785	. . . . 5 ⊢ (𝐶 ∈ 𝐷 → (∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))
10	1, 8, 9	e23 40580	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   [𝐶 / 𝑦][𝐴 / 𝑥]𝜑   )
11	10	in3 40434	. . 3 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ▶   (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)   )
12	11	in2 40430	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))   )
13	12	in1 40396	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))

Syntax hints:   → wi 4   ∈ wcel 2079  ∀wral 3103  [wsbc 3701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-v 3434  df-sbc 3702  df-vd1 40395  df-vd2 40403  df-vd3 40415

This theorem is referenced by: (None)