Theorem vtoclri 1393

Description: Implicit substitution of a class for a set variable.

Hypotheses

Ref	Expression
vtoclri.1	⊢ (x = A → (φ ↔ ψ))
vtoclri.2	⊢ ∀x ∈ B φ

Assertion

Ref	Expression
vtoclri	⊢ (A ∈ B → ψ)

Distinct variable group(s):   x,A   x,B   ψ,x

Proof of Theorem vtoclri

Step	Hyp	Ref	Expression
1		vtoclri.1	. 2 ⊢ (x = A → (φ ↔ ψ))
2		vtoclri.2	. . 3 ⊢ ∀x ∈ B φ
3	2	rspec 1246	. 2 ⊢ (x ∈ B → φ)
4	1, 3	vtoclga 1387	1 ⊢ (A ∈ B → ψ)

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

This theorem is referenced by:  omsdomnn 3424  arch 4521  discrlem 4716  climunii 4883  hlimcaui 5141

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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