Theorem ralbidv2 3182

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 1953	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒)))
3		df-ral 3078	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		df-ral 3078	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒))
5	2, 3, 4	3bitr4g 317	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   ∈ wcel 2145  ∀wral 3077

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-ral 3078

This theorem is used by:  ralbidva  3184  raleqbidv  3335  ralssOLD  4006  oneqmini  6416  ordunisuc2  7855  dfsmo2  8355  wemapsolem  9544  zorn2lem1  10574  raluz  13023  limsupgle  15644  ello12  15683  elo12  15694  lo1resb  15731  rlimresb  15732  o1resb  15733  isprm3  16858  isprm7  16884  ist1-2  23665  hausdiag  23964  xkopt  23974  cnflf  24321  cnfcf  24361  metcnp  24860  caucfil  25604  mdegleb  26382  islinds5  33923  islbs5  33935  eulerpartlemgvv  35008  filnetlem4  37169  mnuunid  45260  iineq12dv  46120  hoidmvle  47609  tmachlem-agreeprod  47946  elbigo2  49663  ralbidb  49909  ralbidc  49910