Theorem ralbii2 3082

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 1826	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
3		df-ral 3055	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
4		df-ral 3055	. 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 1545   ∈ wcel 2119  ∀wral 3054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816

This theorem depends on definitions:  df-bi 208  df-ral 3055

This theorem is referenced by:  ralbiia  3084  ralcom3  3090  raleqbii  3312  ralrab  3642  raldifb  4086  ralin  4184  raldifsni  4735  reusv2  5339  dfsup2  9354  iscard2  9898  acnnum  9972  dfac9  10057  dfacacn  10062  raluz2  12845  ralrp  12962  isprm4  16651  isdomn2OLD  20691  sdrgacs  20780  isnrm2  23348  ismbl  25518  ellimc3  25871  dchrelbas2  27225  onsis  28291  ons2ind  28292  h1dei  31646  iineq1i  36431  ixpeq1i  36435  fnwe2lem2  43503