Theorem cbvrexfw 2706

Description: Rule used to change bound variables, using implicit substitution. Version of cbvrexf 2708 with a disjoint variable condition. Although we don't do so yet, we expect this disjoint variable condition will allow us to remove reliance on ax-i12 1517 and ax-bndl 1519 in the proof. (Contributed by FL, 27-Apr-2008.) (Revised by Gino Giotto, 10-Jan-2024.)

Hypotheses

Ref	Expression
cbvrexfw.1	⊢ Ⅎ𝑥𝐴
cbvrexfw.2	⊢ Ⅎ𝑦𝐴
cbvrexfw.3	⊢ Ⅎ𝑦𝜑
cbvrexfw.4	⊢ Ⅎ𝑥𝜓
cbvrexfw.5	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
cbvrexfw	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem cbvrexfw

Step	Hyp	Ref	Expression
1		cbvrexfw.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
2	1	nfcri 2323	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
3		cbvrexfw.3	. . . 4 ⊢ Ⅎ𝑦𝜑
4	2, 3	nfan 1575	. . 3 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)
5		cbvrexfw.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
6	5	nfcri 2323	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
7		cbvrexfw.4	. . . 4 ⊢ Ⅎ𝑥𝜓
8	6, 7	nfan 1575	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)
9		eleq1w 2248	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10		cbvrexfw.5	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
11	9, 10	anbi12d 473	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ 𝜓)))
12	4, 8, 11	cbvexv1 1762	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
13		df-rex 2471	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
14		df-rex 2471	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
15	12, 13, 14	3bitr4i 212	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  Ⅎwnf 1470  ∃wex 1502   ∈ wcel 2158  Ⅎwnfc 2316  ∃wrex 2466

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471

This theorem is referenced by:  cbvrexw  2710  nnwofdc  12052