Theorem raleqf 3339

Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See raleq 3306 for a version based on fewer axioms. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)

Hypotheses

Ref	Expression
raleqf.1	⊢ Ⅎ𝑥𝐴
raleqf.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
raleqf	⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))

Proof of Theorem raleqf

Step	Hyp	Ref	Expression
1		raleqf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2		raleqf.2	. . . 4 ⊢ Ⅎ𝑥𝐵
3	1, 2	nfeq 2913	. . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		eleq2 2824	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	4	imbi1d 341	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐵 → 𝜑)))
6	3, 5	albid 2223	. 2 ⊢ (𝐴 = 𝐵 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑)))
7		df-ral 3053	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
8		df-ral 3053	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))
9	6, 7, 8	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540   ∈ wcel 2109  Ⅎwnfc 2884  ∀wral 3052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053

This theorem is referenced by:  rexeqf  3340  raleqbid  3342  dfon2lem3  35808  indexa  37762  ralbi12f  38189  iineq12f  38193  ac6s6f  38202  raleqd  45128  stoweidlem28  46024  stoweidlem52  46048  fourierdlem31  46134  fourierdlem68  46170  fourierdlem103  46205  fourierdlem104  46206