Theorem rspsbc2 45108

Description: rspsbc 3835 with two quantifying variables. This proof is rspsbc2VD 45428 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rspsbc2	⊢ (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 rspsbc2

Step	Hyp	Ref	Expression
1		idd 25	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → 𝐶 ∈ 𝐷))
2		rspsbc 3835	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑))
3	2	a1d 26	. . 3 ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑)))
4		sbcralg 3830	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑))
5	4	biimpd 232	. . 3 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑))
6	3, 5	syl6d 76	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑)))
7		rspsbc 3835	. 2 ⊢ (𝐶 ∈ 𝐷 → (∀𝑦 ∈ 𝐷 [𝐴 / 𝑥]𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑))
8	1, 6, 7	syl10 80	1 ⊢ (𝐴 ∈ 𝐵 → (𝐶 ∈ 𝐷 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐷 𝜑 → [𝐶 / 𝑦][𝐴 / 𝑥]𝜑)))

Syntax hints:   → wi 4   ∈ wcel 2145  ∀wral 3079  [wsbc 3747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-v 3459  df-sbc 3748

This theorem is referenced by:  tratrb  45110  tratrbVD  45434