Theorem rexcom4 1361

Description: Commutation of restricted and unrestricted existential quantifiers.

Assertion

Ref	Expression
rexcom4	⊢ (∃x ∈ A ∃yφ ↔ ∃y∃x ∈ A φ)

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

Proof of Theorem rexcom4

Step	Hyp	Ref	Expression
1		rexcom 1313	. 2 ⊢ (∃y ∈ V ∃x ∈ A φ ↔ ∃x ∈ A ∃y ∈ V φ)
2		rexv 1358	. 2 ⊢ (∃y ∈ V ∃x ∈ A φ ↔ ∃y∃x ∈ A φ)
3		rexv 1358	. . 3 ⊢ (∃y ∈ V φ ↔ ∃yφ)
4	3	birex 1224	. 2 ⊢ (∃x ∈ A ∃y ∈ V φ ↔ ∃x ∈ A ∃yφ)
5	1, 2, 4	3bitr3r 157	1 ⊢ (∃x ∈ A ∃yφ ↔ ∃y∃x ∈ A φ)

Syntax hints:   ↔ wb 127  ∃wex 678  ∃wrex 1202  Vcvv 1348

This theorem is referenced by:  uni0b 1939  cnvuni 2521  aceq5lem2 3559

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-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349

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