Theorem ralbii2 3089

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 1821	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
3		df-ral 3062	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
4		df-ral 3062	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
5	2, 3, 4	3bitr4i 302	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1539   ∈ wcel 2106  ∀wral 3061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811

This theorem depends on definitions:  df-bi 206  df-ral 3062

This theorem is referenced by:  ralbiia  3091  ralcom3  3097  raleqbii  3338  ralrab  3689  raldifb  4144  raldifsni  4798  reusv2  5401  dfsup2  9438  iscard2  9970  acnnum  10046  dfac9  10130  dfacacn  10135  raluz2  12880  ralrp  12993  isprm4  16620  sdrgacs  20416  isdomn2  20914  isnrm2  22861  ismbl  25042  ellimc3  25395  dchrelbas2  26737  h1dei  30798  ralin  37110  fnwe2lem2  41783