Theorem vtoclef 1392

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

Hypotheses

Ref	Expression
vtoclef.1	⊢ (φ → ∀xφ)
vtoclef.2	⊢ A ∈ V
vtoclef.3	⊢ (x = A → φ)

Assertion

Ref	Expression
vtoclef	⊢ φ

Distinct variable group(s):   x,A

Proof of Theorem vtoclef

Step	Hyp	Ref	Expression
1		vtoclef.2	. . 3 ⊢ A ∈ V
2	1	isseti 1352	. 2 ⊢ ∃x x = A
3		vtoclef.1	. . 3 ⊢ (φ → ∀xφ)
4		vtoclef.3	. . 3 ⊢ (x = A → φ)
5	3, 4	19.23ai 746	. 2 ⊢ (∃x x = A → φ)
6	2, 5	ax-mp 6	1 ⊢ φ

Syntax hints:   → wi 2  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348

This theorem is referenced by:  elabf 1414  nn0ind 4612

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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