Theorem ralbidv2 3181

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 1940	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒)))
3		df-ral 3077	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		df-ral 3077	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒))
5	2, 3, 4	3bitr4g 316	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1558   ∈ wcel 2142  ∀wral 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930

This theorem depends on definitions:  df-bi 209  df-ral 3077

This theorem is referenced by:  ralbidva  3183  raleqbidv  3336  ralssOLD  4011  oneqmini  6399  ordunisuc2  7824  dfsmo2  8318  wemapsolem  9498  zorn2lem1  10453  raluz  12897  limsupgle  15504  ello12  15543  elo12  15554  lo1resb  15591  rlimresb  15592  o1resb  15593  isprm3  16717  isprm7  16743  ist1-2  23407  hausdiag  23705  xkopt  23715  cnflf  24062  cnfcf  24102  metcnp  24601  caucfil  25345  mdegleb  26124  islinds5  33553  islbs5  33566  eulerpartlemgvv  34673  filnetlem4  36741  mnuunid  44853  iineq12dv  45684  hoidmvle  47174  elbigo2  49174  ralbidb  49421  ralbidc  49422