Theorem cbvralvw 2771

Description: Version of cbvralv 2767 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)

Hypothesis

Ref	Expression
cbvralvw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvralvw	⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)

Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cbvralvw

Step	Hyp	Ref	Expression
1		eleq1w 2292	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
2		cbvralvw.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	imbi12d 234	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑦 ∈ 𝐴 → 𝜓)))
4	3	cbvalvw 1968	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
5		df-ral 2515	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6		df-ral 2515	. 2 ⊢ (∀𝑦 ∈ 𝐴 𝜓 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
7	4, 5, 6	3bitr4i 212	1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1395   ∈ wcel 2202  ∀wral 2510

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-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-clel 2227  df-ral 2515

This theorem is referenced by:  cbvral2vw  2778  cc1  7484  zsupssdc  10499  wrdind  11307  wrd2ind  11308  reuccatpfxs1  11332  prmpwdvds  12946  nninfdclemcl  13087  grpinvalem  13486  grpinva  13487  issubg4m  13798  isnsg2  13808  elnmz  13813  fsumdvdsmul  15734  2sqlem6  15868  2sqlem10  15873  uspgr2wlkeq  16235  depindlem1  16376  bj-charfunbi  16457