Theorem cbvralvw 2733

Description: Version of cbvralv 2729 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 2257	. . . 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 1934	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝜓))
5		df-ral 2480	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑))
6		df-ral 2480	. 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 1362   ∈ wcel 2167  ∀wral 2475

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-clel 2192  df-ral 2480

This theorem is referenced by:  cbvral2vw  2740  cc1  7332  zsupssdc  10328  prmpwdvds  12524  nninfdclemcl  12665  grpinvalem  13028  grpinva  13029  issubg4m  13323  isnsg2  13333  elnmz  13338  fsumdvdsmul  15227  2sqlem6  15361  2sqlem10  15366  bj-charfunbi  15457