Theorem raleqf 3345

Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. See raleq 3319 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 2939	. . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		eleq2 2853	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	4	imbi1d 343	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐵 → 𝜑)))
6	3, 5	albid 2259	. 2 ⊢ (𝐴 = 𝐵 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑)))
7		df-ral 3079	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
8		df-ral 3079	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))
9	6, 7, 8	3bitr4g 316	1 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1560   = wceq 1562   ∈ wcel 2144  Ⅎwnfc 2911  ∀wral 3078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-nf 1806  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079

This theorem is referenced by:  rexeqf  3346  raleqbid  3347  dfon2lem3  36138  indexa  38237  ralbi12f  38664  iineq12f  38668  ac6s6f  38677  raleqd  45720  stoweidlem28  46607  stoweidlem52  46631  fourierdlem31  46717  fourierdlem68  46753  fourierdlem103  46788  fourierdlem104  46789