Theorem ralbii2 3130

Description: Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.)

Hypothesis

Ref	Expression
ralbii2.1	⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐵 → 𝜓))

Assertion

Ref	Expression
ralbii2	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)

Proof of Theorem ralbii2

Step	Hyp	Ref	Expression
1		ralbii2.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐵 → 𝜓))
2	1	albii 1801	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
3		df-ral 3110	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
4		df-ral 3110	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
5	2, 3, 4	3bitr4i 304	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1520   ∈ wcel 2081  ∀wral 3105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791

This theorem depends on definitions:  df-bi 208  df-ral 3110

This theorem is referenced by:  ralbiia  3131  ralbii  3132  raleqbii  3198  ralrab  3623  raldifb  4042  raldifsni  4635  reusv2  5195  dfsup2  8754  iscard2  9251  acnnum  9324  dfac9  9408  dfacacn  9413  raluz2  12146  ralrp  12259  isprm4  15857  sdrgacs  19270  isdomn2  19761  isnrm2  21650  ismbl  23810  ellimc3  24160  dchrelbas2  25495  h1dei  29018  fnwe2lem2  39136