Theorem sbc6 1453

Description: An equivalence for class substitution.

Hypothesis

Ref	Expression
sbc6.1	⊢ A ∈ V

Assertion

Ref	Expression
sbc6	⊢ ([A / x]φ ↔ ∀x(x = A → φ))

Distinct variable group(s):   x,A

Proof of Theorem sbc6

Step	Hyp	Ref	Expression
1		sbc6.1	. . . 4 ⊢ A ∈ V
2	1	hbsbcv 1447	. . 3 ⊢ ([A / x]φ → ∀x[A / x]φ)
3		sbceq1 1443	. . 3 ⊢ (x = A → (φ ↔ [A / x]φ))
4	2, 1, 3	ceqsal 1363	. 2 ⊢ (∀x(x = A → φ) ↔ [A / x]φ)
5	4	bicomi 150	1 ⊢ ([A / x]φ ↔ ∀x(x = A → φ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = wceq 1091   ∈ wcel 1092  Vcvv 1348  [wsbc 1440

This theorem is referenced by:  sbcie 1455

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-10 800  ax-11 801  ax-12 802  ax-16 922  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  df-sbc 1441

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