Theorem hbsbcv 1447

Description: Bound variable hypothesis builder for class substitution.

Hypothesis

Ref	Expression
hbsbcv.1	⊢ A ∈ V

Assertion

Ref	Expression
hbsbcv	⊢ ([A / x]φ → ∀x[A / x]φ)

Distinct variable group(s):   x,A

Proof of Theorem hbsbcv

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 ⊢ (y ∈ A → ∀x y ∈ A)
2	1	hbsbc 1446	. 2 ⊢ ((A ∈ V → [A / x]φ) → ∀x(A ∈ V → [A / x]φ))
3		hbsbcv.1	. . 3 ⊢ A ∈ V
4	3	a1bi 172	. 2 ⊢ ([A / x]φ ↔ (A ∈ V → [A / x]φ))
5	4	bial 695	. 2 ⊢ (∀x[A / x]φ ↔ ∀x(A ∈ V → [A / x]φ))
6	2, 4, 5	3imtr4 192	1 ⊢ ([A / x]φ → ∀x[A / x]φ)

Syntax hints:   → wi 2  ∀wal 672   ∈ wcel 1092  Vcvv 1348  [wsbc 1440

This theorem is referenced by:  sbc6 1453  findes 2400  tfindes 2404  nn1suc 4435  uzind 4603  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-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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