Theorem ralbidv2 3148

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Apr-1997.)

Hypothesis

Ref	Expression
ralbidv2.1	⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒)))

Assertion

Ref	Expression
ralbidv2	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem ralbidv2

Step	Hyp	Ref	Expression
1		ralbidv2.1	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒)))
2	1	albidv 1920	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒)))
3		df-ral 3045	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		df-ral 3045	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒))
5	2, 3, 4	3bitr4g 314	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   ∈ wcel 2109  ∀wral 3044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910

This theorem depends on definitions:  df-bi 207  df-ral 3045

This theorem is referenced by:  ralbidva  3150  raleqbidv  3309  ralssOLD  4012  oneqmini  6360  ordunisuc2  7777  dfsmo2  8270  wemapsolem  9442  zorn2lem1  10390  raluz  12797  limsupgle  15384  ello12  15423  elo12  15434  lo1resb  15471  rlimresb  15472  o1resb  15473  isprm3  16594  isprm7  16619  ist1-2  23232  hausdiag  23530  xkopt  23540  cnflf  23887  cnfcf  23927  metcnp  24427  caucfil  25181  mdegleb  25967  islinds5  33304  islbs5  33317  eulerpartlemgvv  34344  filnetlem4  36359  mnuunid  44254  iineq12dv  45088  hoidmvle  46585  elbigo2  48541  ralbidb  48788  ralbidc  48789