Theorem vtocleg 1390

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

Hypothesis

Ref	Expression
vtocleg.1	⊢ (x = A → φ)

Assertion

Ref	Expression
vtocleg	⊢ (A ∈ B → φ)

Distinct variable group(s):   x,A   φ,x

Proof of Theorem vtocleg

Step	Hyp	Ref	Expression
1		elex 1356	. 2 ⊢ (A ∈ B → ∃x x = A)
2		vtocleg.1	. . 3 ⊢ (x = A → φ)
3	2	19.23aiv 952	. 2 ⊢ (∃x x = A → φ)
4	1, 3	syl 12	1 ⊢ (A ∈ B → φ)

Syntax hints:   → wi 2  ∃wex 678   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  vtocle 1391  a4sbc 1444  hbsbcg 1445  noel 1711  prex 1892

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-gen 677  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-v 1349

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