Theorem raleqf 2745

Description: Equality theorem for restricted universal quantifier, with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by NM, 7-Mar-2004.) (Revised by Andrew Salmon, 11-Jul-2011.)

Hypotheses

Ref	Expression
raleq1f.1	⊢ Ⅎ𝑥𝐴
raleq1f.2	⊢ Ⅎ𝑥𝐵

Assertion

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

Proof of Theorem raleqf

Step	Hyp	Ref	Expression
1		raleq1f.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2		raleq1f.2	. . . 4 ⊢ Ⅎ𝑥𝐵
3	1, 2	nfeq 2400	. . 3 ⊢ Ⅎ𝑥 𝐴 = 𝐵
4		eleq2 2302	. . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
5	4	imbi1d 231	. . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐵 → 𝜑)))
6	3, 5	albid 1668	. 2 ⊢ (𝐴 = 𝐵 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑)))
7		df-ral 2533	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
8		df-ral 2533	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜑))
9	6, 7, 8	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1400   = wceq 1402   ∈ wcel 2209  Ⅎwnfc 2379  ∀wral 2528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533

This theorem is referenced by:  raleq  2749  repizf2  4297  ellimc3apf  15687