Theorem ralcom4 2576

Description: Commutation of restricted and unrestricted universal quantifiers. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Assertion

Ref	Expression
ralcom4	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

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

Proof of Theorem ralcom4

Step	Hyp	Ref	Expression
1		ralcom 2473	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ V 𝜑 ↔ ∀𝑦 ∈ V ∀𝑥 ∈ 𝐴 𝜑)
2		ralv 2571	. . 3 ⊢ (∀𝑦 ∈ V 𝜑 ↔ ∀𝑦𝜑)
3	2	ralbii 2330	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ V 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦𝜑)
4		ralv 2571	. 2 ⊢ (∀𝑦 ∈ V ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)
5	1, 3, 4	3bitr3i 199	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦𝜑 ↔ ∀𝑦∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 98  ∀wal 1241  ∀wral 2306  Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559

This theorem is referenced by:  uniiunlem  3028  uni0b  3605  iunss  3698  trint  3869  reliun  4458  funimass4  5224  ralrnmpt2  5615