Theorem ralbidv2 3152

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  3154  raleqbidv  3319  ralssOLD  4023  oneqmini  6385  ordunisuc2  7820  dfsmo2  8316  wemapsolem  9503  zorn2lem1  10449  raluz  12855  limsupgle  15443  ello12  15482  elo12  15493  lo1resb  15530  rlimresb  15531  o1resb  15532  isprm3  16653  isprm7  16678  ist1-2  23234  hausdiag  23532  xkopt  23542  cnflf  23889  cnfcf  23929  metcnp  24429  caucfil  25183  mdegleb  25969  islinds5  33338  islbs5  33351  eulerpartlemgvv  34367  filnetlem4  36369  mnuunid  44266  iineq12dv  45100  hoidmvle  46598  elbigo2  48541  ralbidb  48788  ralbidc  48789