Theorem rspsbc2VD 45931

Description: Virtual deduction proof of rspsbc2 45611. 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 45824	⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   [𝐴 / 𝑥]∀𝑦 ∈ 𝐷𝜑   )
5:1,4,?: e13 45824	⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷𝜑   ▶   ∀𝑦 ∈ 𝐷[𝐴 / 𝑥]𝜑   )
6:2,5,?: e23 45831	⊢ (   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 45690	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ▶   𝐶 ∈ 𝐷   )
2		idn1 45651	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ▶   𝐴 ∈ 𝐵   )
3		idn3 45692	. . . . . . 7 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   )
4		rspsbc 3825	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑))
5	2, 3, 4	e13 45824	. . . . . 6 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   [𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑   )
6		sbcralg 3820	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑))
7	6	biimpd 232	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑))
8	2, 5, 7	e13 45824	. . . . 5 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑   )
9		rspsbc 3825	. . . . 5 ⊢ (𝐶 ∈ 𝐷 → (∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))
10	1, 8, 9	e23 45831	. . . 4 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ,   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑   ▶   [𝐶 / 𝑦][𝐴 / 𝑥]𝜑   )
11	10	in3 45686	. . 3 ⊢ (   𝐴 ∈ 𝐵   ,   𝐶 ∈ 𝐷   ▶   (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)   )
12	11	in2 45682	. 2 ⊢ (   𝐴 ∈ 𝐵   ▶   (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))   )
13	12	in1 45648	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  ∀wral 3076  [wsbc 3738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-v 3452  df-sbc 3739  df-vd1 45647  df-vd2 45655  df-vd3 45667

This theorem is used by: (None)