Theorem hbsbcg 1445

Description: Bound variable hypothesis builder for class substitution.

Hypothesis

Ref	Expression
hbsbcg.1	⊢ (y ∈ A → ∀x y ∈ A)

Assertion

Ref	Expression
hbsbcg	⊢ (A ∈ B → ([A / x]φ → ∀x[A / x]φ))

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

Proof of Theorem hbsbcg

Step	Hyp	Ref	Expression
1		dfsbcq 1442	. . 3 ⊢ (z = A → ([z / x]φ ↔ [A / x]φ))
2		ax-17 925	. . . . . 6 ⊢ (y ∈ z → ∀x y ∈ z)
3		hbsbcg.1	. . . . . 6 ⊢ (y ∈ A → ∀x y ∈ A)
4	2, 3	hbeq 1171	. . . . 5 ⊢ (z = A → ∀x z = A)
5	4, 1	biald 782	. . . 4 ⊢ (z = A → (∀x[z / x]φ ↔ ∀x[A / x]φ))
6		hbs1 986	. . . 4 ⊢ ([z / x]φ → ∀x[z / x]φ)
7	5, 6	syl5bi 183	. . 3 ⊢ (z = A → ([z / x]φ → ∀x[A / x]φ))
8	1, 7	sylbird 180	. 2 ⊢ (z = A → ([A / x]φ → ∀x[A / x]φ))
9	8	vtocleg 1390	1 ⊢ (A ∈ B → ([A / x]φ → ∀x[A / x]φ))

Syntax hints:   → wi 2  ∀wal 672   ∈ wel 803  [wsb 852   = wceq 1091   ∈ wcel 1092  [wsbc 1440

This theorem is referenced by:  hbsbc 1446

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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