Theorem rspc2dv 3594

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 3591	. 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 2147  ∀wral 3080

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081

This theorem is used by:  prmidlprop  21683  mulscom  28582  addsdilem3  28596  addsdilem4  28597  mulsasslem3  28608  edomediv  33955  rprmdvds  34155  mplvrpmga  34288  vieta  34323  cvxsconn  36216  nmulprop  37154  nmulcom  37158  nadddilem1  37184  nadddilem3  37186  oppcmndclem  50462  ssccatid  50517  termcbasmo  50949  fulltermc2  50978  arweuthinc  50995  arweutermc  50996