Theorem sbcralg 1991

Description: Interchange class substitution and restricted quantifier.

Assertion

Ref	Expression
sbcralg	⊢ (A ∈ C → ([A / x]∀y ∈ B φ ↔ ∀y ∈ B [A / x]φ))

Distinct variable groups:   y,A   x,B   x,y

Proof of Theorem sbcralg

Step	Hyp	Ref	Expression
1		elisset 1814	. 2 ⊢ (A ∈ C → A ∈ V)
2		ax-17 970	. . . 4 ⊢ (z ∈ A → ∀y z ∈ A)
3	2	ax-gen 962	. . 3 ⊢ ∀z(z ∈ A → ∀y z ∈ A)
4		ax-17 970	. . . . 5 ⊢ (A ∈ V → ∀y A ∈ V)
5	3	hbth 1000	. . . . 5 ⊢ (∀z(z ∈ A → ∀y z ∈ A) → ∀y∀z(z ∈ A → ∀y z ∈ A))
6	4, 5	hban 1008	. . . 4 ⊢ ((A ∈ V ⋀ ∀z(z ∈ A → ∀y z ∈ A)) → ∀y(A ∈ V ⋀ ∀z(z ∈ A → ∀y z ∈ A)))
7		sbcralt 1987	. . . 4 ⊢ (∀y(A ∈ V ⋀ ∀z(z ∈ A → ∀y z ∈ A)) → ([A / x]∀y ∈ B φ ↔ ∀y ∈ B [A / x]φ))
8	6, 7	syl 10	. . 3 ⊢ ((A ∈ V ⋀ ∀z(z ∈ A → ∀y z ∈ A)) → ([A / x]∀y ∈ B φ ↔ ∀y ∈ B [A / x]φ))
9	3, 8	mpan2 695	. 2 ⊢ (A ∈ V → ([A / x]∀y ∈ B φ ↔ ∀y ∈ B [A / x]φ))
10	1, 9	syl 10	1 ⊢ (A ∈ C → ([A / x]∀y ∈ B φ ↔ ∀y ∈ B [A / x]φ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 953   ∈ wcel 957  [wsbc 1169  ∀wral 1643  Vcvv 1808

This theorem is referenced by:  r19.12sn 2441  csbfsum 6980

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-ral 1647  df-v 1809  df-sbc 1939

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