Theorem rspc2dv 3591

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 3588	. 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 2145  ∀wral 3077

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 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078

This theorem is used by:  prmidlprop  21632  mulscom  28525  addsdilem3  28539  addsdilem4  28540  mulsasslem3  28551  rprmdvds  34051  mplvrpmga  34177  vieta  34212  cvxsconn  36008  nmulprop  36939  nmulcom  36943  nadddilem1  36969  nadddilem3  36971  oppcmndclem  50124  ssccatid  50179  termcbasmo  50590  fulltermc2  50619  arweuthinc  50636  arweutermc  50637