Theorem rcla42v 1404

Description: 2-variable restricted specialization with implicit substitution.

Hypotheses

Ref	Expression
rcla42v.1	⊢ (x = A → (φ ↔ χ))
rcla42v.2	⊢ (y = B → (χ ↔ ψ))

Assertion

Ref	Expression
rcla42v	⊢ (∀x ∈ C ∀y ∈ D φ → ((A ∈ C ∧ B ∈ D) → ψ))

Distinct variable group(s):   x,y,A   x,C   x,D   y,B   y,D   χ,x   ψ,y

Proof of Theorem rcla42v

Step	Hyp	Ref	Expression
1		rcla42v.1	. . . 4 ⊢ (x = A → (φ ↔ χ))
2	1	biraldv 1219	. . 3 ⊢ (x = A → (∀y ∈ D φ ↔ ∀y ∈ D χ))
3	2	rcla4v 1402	. 2 ⊢ (∀x ∈ C ∀y ∈ D φ → (A ∈ C → ∀y ∈ D χ))
4		rcla42v.2	. . . . 5 ⊢ (y = B → (χ ↔ ψ))
5	4	rcla4v 1402	. . . 4 ⊢ (∀y ∈ D χ → (B ∈ D → ψ))
6	5	syl3 18	. . 3 ⊢ ((A ∈ C → ∀y ∈ D χ) → (A ∈ C → (B ∈ D → ψ)))
7	6	imp3a 279	. 2 ⊢ ((A ∈ C → ∀y ∈ D χ) → ((A ∈ C ∧ B ∈ D) → ψ))
8	3, 7	syl 12	1 ⊢ (∀x ∈ C ∀y ∈ D φ → ((A ∈ C ∧ B ∈ D) → ψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201

This theorem is referenced by:  isorel 2932  isocnv 2934  isotr 2935  isotrALT 2936  fiint 3445  seqrn 4673  infxpidmlem7 4939  shaddclt 5123  shmulclt 5124  stjt 5676  stcltr1 5707

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349

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