Theorem rexxpf 5718

Description: Version of rexxp 5713 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
ralxpf.1	⊢ Ⅎ𝑦𝜑
ralxpf.2	⊢ Ⅎ𝑧𝜑
ralxpf.3	⊢ Ⅎ𝑥𝜓
ralxpf.4	⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rexxpf	⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)

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

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

Proof of Theorem rexxpf

Step	Hyp	Ref	Expression
1		ralxpf.1	. . . . . 6 ⊢ Ⅎ𝑦𝜑
2	1	nfn 1857	. . . . 5 ⊢ Ⅎ𝑦 ¬ 𝜑
3		ralxpf.2	. . . . . 6 ⊢ Ⅎ𝑧𝜑
4	3	nfn 1857	. . . . 5 ⊢ Ⅎ𝑧 ¬ 𝜑
5		ralxpf.3	. . . . . 6 ⊢ Ⅎ𝑥𝜓
6	5	nfn 1857	. . . . 5 ⊢ Ⅎ𝑥 ¬ 𝜓
7		ralxpf.4	. . . . . 6 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))
8	7	notbid 320	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (¬ 𝜑 ↔ ¬ 𝜓))
9	2, 4, 6, 8	ralxpf 5717	. . . 4 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓)
10		ralnex 3236	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ¬ ∃𝑧 ∈ 𝐵 𝜓)
11	10	ralbii 3165	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓)
12	9, 11	bitri 277	. . 3 ⊢ (∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓)
13	12	notbii 322	. 2 ⊢ (¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓)
14		dfrex2 3239	. 2 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ¬ ∀𝑥 ∈ (𝐴 × 𝐵) ¬ 𝜑)
15		dfrex2 3239	. 2 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ ∃𝑧 ∈ 𝐵 𝜓)
16	13, 14, 15	3bitr4i 305	1 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   = wceq 1537  Ⅎwnf 1784  ∀wral 3138  ∃wrex 3139  ⟨cop 4573   × cxp 5553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-iun 4921  df-opab 5129  df-xp 5561  df-rel 5562

This theorem is referenced by:  iunxpf  5719  wdom2d2  39652