Theorem ralcom 2771

Description: Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.)

Assertion

Ref	Expression
ralcom	⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀y ∈ B ∀x ∈ A φ)

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

Allowed substitution hints:   φ(x,y)   A(x)   B(y)

Proof of Theorem ralcom

Step	Hyp	Ref	Expression
1		nfcv 2489	. 2 ⊢ ℲyA
2		nfcv 2489	. 2 ⊢ ℲxB
3	1, 2	ralcomf 2769	1 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀y ∈ B ∀x ∈ A φ)

Syntax hints:   ↔ wb 176  ∀wral 2614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619

This theorem is referenced by:  ralcom4  2877  ssint  3942  iinrab2  4029  fununi  5160  isocnv2  5492