Theorem cbvrex2 39822

Description: Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 3156. (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,𝐴   𝑧,𝐴   𝑥,𝑦,𝐵   𝑦,𝑧,𝐵   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 2751	. . . 4 ⊢ Ⅎ𝑧𝐵
2		cbvral2.1	. . . 4 ⊢ Ⅎ𝑧𝜑
3	1, 2	nfrex 2990	. . 3 ⊢ Ⅎ𝑧∃𝑦 ∈ 𝐵 𝜑
4		nfcv 2751	. . . 4 ⊢ Ⅎ𝑥𝐵
5		cbvral2.2	. . . 4 ⊢ Ⅎ𝑥𝜒
6	4, 5	nfrex 2990	. . 3 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐵 𝜒
7		cbvral2.5	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜒))
8	7	rexbidv 3034	. . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜒))
9	3, 6, 8	cbvrex 3144	. 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	cbvrex 3144	. . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑤 ∈ 𝐵 𝜓)
14	13	rexbii 3023	. 2 ⊢ (∃𝑧 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)
15	9, 14	bitri 263	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 𝜓)

Syntax hints:   → wi 4   ↔ wb 195  Ⅎwnf 1699  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902

This theorem is referenced by: (None)