Theorem ralbidv2 3013

Description: Formula-building rule for restricted universal quantifier (deduction rule). (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 1889	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒)))
3		df-ral 2946	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		df-ral 2946	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒))
5	2, 3, 4	3bitr4g 303	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1521   ∈ wcel 2030  ∀wral 2941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879

This theorem depends on definitions:  df-bi 197  df-ral 2946

This theorem is referenced by:  ralbidva  3014  ralss  3701  oneqmini  5814  ordunisuc2  7086  dfsmo2  7489  wemapsolem  8496  zorn2lem1  9356  raluz  11774  limsupgle  14252  ello12  14291  elo12  14302  lo1resb  14339  rlimresb  14340  o1resb  14341  isprm3  15443  isprm7  15467  ist1  21173  ist1-2  21199  hausdiag  21496  xkopt  21506  cnflf  21853  cnfcf  21893  metcnp  22393  caucfil  23127  mdegleb  23869  eulerpartlemgvv  30566  filnetlem4  32501  hoidmvle  41135  elbigo2  42671