Theorem hbsbc 1446

Description: Bound variable hypothesis builder for class substitution. (The antecedent ensures that A is a set, which is necessary if we restrict ourselves to using only the "weak" class substitution definition dfsbcq 1442.)

Hypothesis

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

Assertion

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

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

Proof of Theorem hbsbc

Step	Hyp	Ref	Expression
1		hbsbcg.1	. . 3 ⊢ (y ∈ A → ∀x y ∈ A)
2		ax-17 925	. . 3 ⊢ (y ∈ B → ∀x y ∈ B)
3	1, 2	hbel 1172	. 2 ⊢ (A ∈ B → ∀x A ∈ B)
4	1	hbsbcg 1445	. 2 ⊢ (A ∈ B → ([A / x]φ → ∀x[A / x]φ))
5	3, 4	hbim1 781	1 ⊢ ((A ∈ B → [A / x]φ) → ∀x(A ∈ B → [A / x]φ))

Syntax hints:   → wi 2  ∀wal 672   ∈ wcel 1092  [wsbc 1440

This theorem is referenced by:  hbsbcv 1447  sbc5g 1450  sÙc6g 1451  elrabsf 1456  sbcel1 1466  sbcel2 1467  reuuni4 1959  nn1suc 4435

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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