Theorem hbrex 1238

Description: Bound-variable hypothesis builder for restricted quantification.

Hypotheses

Ref	Expression
hbrex.1	⊢ (y ∈ A → ∀x y ∈ A)
hbrex.2	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
hbrex	⊢ (∃y ∈ A φ → ∀x∃y ∈ A φ)

Distinct variable group(s):   x,y

Proof of Theorem hbrex

Step	Hyp	Ref	Expression
1		hbrex.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
2		hbrex.2	. . . 4 ⊢ (φ → ∀xφ)
3	1, 2	hban 704	. . 3 ⊢ ((y ∈ A ∧ φ) → ∀x(y ∈ A ∧ φ))
4	3	hbex 701	. 2 ⊢ (∃y(y ∈ A ∧ φ) → ∀x∃y(y ∈ A ∧ φ))
5		df-rex 1206	. 2 ⊢ (∃y ∈ A φ ↔ ∃y(y ∈ A ∧ φ))
6	5	bial 695	. 2 ⊢ (∀x∃y ∈ A φ ↔ ∀x∃y(y ∈ A ∧ φ))
7	4, 5, 6	3imtr4 192	1 ⊢ (∃y ∈ A φ → ∀x∃y ∈ A φ)

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wcel 1092  ∃wrex 1202

This theorem is referenced by:  r19.12 1281  iunrab 2022  abrexexlem2 2911  abrexex2 2915  hbrdg 2974  elrnoprab 3054

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

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-rex 1206

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