Theorem rspc2dv 3637

Description: 2-variable restricted specialization, using implicit substitution. (Contributed by Scott Fenton, 6-Mar-2025.)

Hypotheses

Ref	Expression
rspc2dv.1	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
rspc2dv.2	⊢ (𝑦 = 𝐵 → (𝜃 ↔ 𝜒))
rspc2dv.3	⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜓)
rspc2dv.4	⊢ (𝜑 → 𝐴 ∈ 𝐶)
rspc2dv.5	⊢ (𝜑 → 𝐵 ∈ 𝐷)

Assertion

Ref	Expression
rspc2dv	⊢ (𝜑 → 𝜒)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶   𝑥,𝐷,𝑦   𝜒,𝑦   𝜃,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥)   𝜃(𝑦)   𝐵(𝑥)   𝐶(𝑦)

Proof of Theorem rspc2dv

Step	Hyp	Ref	Expression
1		rspc2dv.4	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
2		rspc2dv.5	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
3		rspc2dv.3	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜓)
4		rspc2dv.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
5		rspc2dv.2	. . 3 ⊢ (𝑦 = 𝐵 → (𝜃 ↔ 𝜒))
6	4, 5	rspc2va 3634	. 2 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜓) → 𝜒)
7	1, 2, 3, 6	syl21anc 838	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106  ∀wral 3059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060

This theorem is referenced by:  mulscom  28180  addsdilem3  28194  addsdilem4  28195  mulsasslem3  28206  rprmdvds  33527  cvxsconn  35228