Theorem hblem 1170

Description: Lemma for hbeq 1171 and hbel 1172.

Hypothesis

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

Assertion

Ref	Expression
hblem	⊢ (z ∈ A → ∀x z ∈ A)

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

Proof of Theorem hblem

Step	Hyp	Ref	Expression
1		eleq1 1149	. . 3 ⊢ (y = z → (y ∈ A ↔ z ∈ A))
2	1	bialdv 935	. . 3 ⊢ (y = z → (∀x y ∈ A ↔ ∀x z ∈ A))
3	1, 2	imbi12d 474	. 2 ⊢ (y = z → ((y ∈ A → ∀x y ∈ A) ↔ (z ∈ A → ∀x z ∈ A)))
4		hblem.1	. 2 ⊢ (y ∈ A → ∀x y ∈ A)
5	3, 4	chv 984	1 ⊢ (z ∈ A → ∀x z ∈ A)

Syntax hints:   → wi 2  ∀wal 672   = weq 797   ∈ wcel 1092

This theorem is referenced by:  hbeq 1171  hbel 1172

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-gen 677  ax-9 799  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099

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