Theorem ralxpxfr2d 3587

Description: Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)

Hypotheses

Ref	Expression
ralxpxfr2d.a	⊢ 𝐴 ∈ V
ralxpxfr2d.b	⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴))
ralxpxfr2d.c	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
ralxpxfr2d	⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 𝜒))

Distinct variable groups:   𝜑,𝑥,𝑧   𝜑,𝑦,𝑥   𝜓,𝑦   𝜓,𝑧   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝜒,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦,𝑧)   𝐴(𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑦,𝑧)   𝐷(𝑦,𝑧)

Proof of Theorem ralxpxfr2d

Step	Hyp	Ref	Expression
1		df-ral 3111	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
2		ralxpxfr2d.b	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴))
3	2	imbi1d 345	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 → 𝜓) ↔ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓)))
4	3	albidv 1921	. . . 4 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜓) ↔ ∀𝑥(∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓)))
5	1, 4	syl5bb 286	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓)))
6		ralcom4 3198	. . . 4 ⊢ (∀𝑦 ∈ 𝐶 ∀𝑥∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓) ↔ ∀𝑥∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
7		ralcom4 3198	. . . . 5 ⊢ (∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ ∀𝑥∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
8	7	ralbii 3133	. . . 4 ⊢ (∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 ∀𝑥∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
9		r19.23v 3238	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓) ↔ (∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓))
10	9	ralbii 3133	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 (∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓))
11		r19.23v 3238	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐶 (∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓) ↔ (∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓))
12	10, 11	bitr2i 279	. . . . 5 ⊢ ((∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
13	12	albii 1821	. . . 4 ⊢ (∀𝑥(∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓) ↔ ∀𝑥∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 (𝑥 = 𝐴 → 𝜓))
14	6, 8, 13	3bitr4ri 307	. . 3 ⊢ (∀𝑥(∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓))
15	5, 14	syl6bb 290	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓)))
16		ralxpxfr2d.c	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒))
17	16	pm5.74da 803	. . . . 5 ⊢ (𝜑 → ((𝑥 = 𝐴 → 𝜓) ↔ (𝑥 = 𝐴 → 𝜒)))
18	17	albidv 1921	. . . 4 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 = 𝐴 → 𝜒)))
19		ralxpxfr2d.a	. . . . 5 ⊢ 𝐴 ∈ V
20		biidd 265	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜒 ↔ 𝜒))
21	19, 20	ceqsalv 3479	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → 𝜒) ↔ 𝜒)
22	18, 21	syl6bb 290	. . 3 ⊢ (𝜑 → (∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ 𝜒))
23	22	2ralbidv 3164	. 2 ⊢ (𝜑 → (∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 ∀𝑥(𝑥 = 𝐴 → 𝜓) ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 𝜒))
24	15, 23	bitrd 282	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑦 ∈ 𝐶 ∀𝑧 ∈ 𝐷 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-nf 1786  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112

This theorem is referenced by:  ralxpmap  8443