Theorem cbvrex2 48268

Description: Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3354. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Hypotheses

Ref	Expression
cbvral2.1	⊢ Ⅎ𝑧𝜑
cbvral2.2	⊢ Ⅎ𝑥𝜒
cbvral2.3	⊢ Ⅎ𝑤𝜒
cbvral2.4	⊢ Ⅎ𝑦𝜓
cbvral2.5	⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))
cbvral2.6	⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))

Assertion

Ref	Expression
cbvrex2	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)

Distinct variable groups:   𝑥,𝑧,𝐴   𝑥,𝑦,𝐵,𝑧   𝑦,𝑤,𝐵

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

Proof of Theorem cbvrex2

Step	Hyp	Ref	Expression
1		nfcv 2922	. . . 4 ⊢ Ⅎ𝑧𝐵
2		cbvral2.1	. . . 4 ⊢ Ⅎ𝑧𝜑
3	1, 2	nfrexw 3310	. . 3 ⊢ Ⅎ𝑧∃𝑦 ∈ 𝐵 𝜑
4		nfcv 2922	. . . 4 ⊢ Ⅎ𝑥𝐵
5		cbvral2.2	. . . 4 ⊢ Ⅎ𝑥𝜒
6	4, 5	nfrexw 3310	. . 3 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐵 𝜒
7		cbvral2.5	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))
8	7	rexbidv 3186	. . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜒))
9	3, 6, 8	cbvrexw 3305	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒)
10		cbvral2.3	. . . 4 ⊢ Ⅎ𝑤𝜒
11		cbvral2.4	. . . 4 ⊢ Ⅎ𝑦𝜓
12		cbvral2.6	. . . 4 ⊢ (𝑦 = 𝑤 → (𝜒 ↔ 𝜓))
13	10, 11, 12	cbvrexw 3305	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑤 ∈ 𝐵 𝜓)
14	13	rexbii 3109	. 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)
15	9, 14	bitri 278	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)

This proof depends on syntax axioms:   ↔ wb 209  Ⅎwnf 1815  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087

This theorem is used by: (None)