Theorem rspc2dv 3598

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 3595	. 2 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜓) → 𝜒)
7	1, 2, 3, 6	syl21anc 851	1 ⊢ (𝜑 → 𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146  ∀wral 3081

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082

This theorem is used by:  prmidlprop  21528  mulscom  28385  addsdilem3  28399  addsdilem4  28400  mulsasslem3  28411  rprmdvds  33875  mplvrpmga  34001  vieta  34036  cvxsconn  35774  nmulprop  36721  nmulcom  36725  nadddilem1  36751  nadddilem3  36753  oppcmndclem  49854  ssccatid  49909  termcbasmo  50320  fulltermc2  50349  arweuthinc  50366  arweutermc  50367