Theorem cleqf 1557

Description: Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions.

Hypotheses

Ref	Expression
cleqf.1	⊢ (y ∈ A → ∀x y ∈ A)
cleqf.2	⊢ (y ∈ B → ∀x y ∈ B)

Assertion

Ref	Expression
cleqf	⊢ (A = B ↔ ∀x(x ∈ A ↔ x ∈ B))

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

Proof of Theorem cleqf

Step	Hyp	Ref	Expression
1		dfcleq 1468	. 2 ⊢ (A = B ↔ ∀y(y ∈ A ↔ y ∈ B))
2		ax-17 969	. . 3 ⊢ ((x ∈ A ↔ x ∈ B) → ∀y(x ∈ A ↔ x ∈ B))
3		cleqf.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
4		cleqf.2	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
5	3, 4	hbbi 1008	. . 3 ⊢ ((y ∈ A ↔ y ∈ B) → ∀x(y ∈ A ↔ y ∈ B))
6		eleq1 1531	. . . 4 ⊢ (x = y → (x ∈ A ↔ y ∈ A))
7		eleq1 1531	. . . 4 ⊢ (x = y → (x ∈ B ↔ y ∈ B))
8	6, 7	bibi12d 628	. . 3 ⊢ (x = y → ((x ∈ A ↔ x ∈ B) ↔ (y ∈ A ↔ y ∈ B)))
9	2, 5, 8	cbval 1163	. 2 ⊢ (∀x(x ∈ A ↔ x ∈ B) ↔ ∀y(y ∈ A ↔ y ∈ B))
10	1, 9	bitr4 176	1 ⊢ (A = B ↔ ∀x(x ∈ A ↔ x ∈ B))

Syntax hints:   → wi 3   ↔ wb 146  ∀wal 952   = wceq 954   ∈ wcel 956

This theorem is referenced by:  abeq2 1565  eq2ab 1570  cbvab 1904  ne0f 2283

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-cleq 1467  df-clel 1470

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