Theorem ralbii2 3072

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 1819	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
3		df-ral 3046	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
4		df-ral 3046	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
5	2, 3, 4	3bitr4i 303	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   ∈ wcel 2109  ∀wral 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 207  df-ral 3046

This theorem is referenced by:  ralbiia  3074  ralcom3  3080  raleqbii  3319  ralrab  3668  raldifb  4115  ralin  4215  raldifsni  4762  reusv2  5361  dfsup2  9402  iscard2  9936  acnnum  10012  dfac9  10097  dfacacn  10102  raluz2  12863  ralrp  12980  isprm4  16661  isdomn2OLD  20628  sdrgacs  20717  isnrm2  23252  ismbl  25434  ellimc3  25787  dchrelbas2  27155  onsis  28179  h1dei  31486  iineq1i  36191  ixpeq1i  36195  fnwe2lem2  43047