Theorem ceqsrex2v 2676

Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.)

Hypotheses

Ref	Expression
ceqsrex2v.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
ceqsrex2v.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
ceqsrex2v	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜒))

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝜒(𝑥)   𝐶(𝑦)

Proof of Theorem ceqsrex2v

Step	Hyp	Ref	Expression
1		anass 381	. . . . . 6 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
2	1	rexbii 2331	. . . . 5 ⊢ (∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐷 (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)))
3		r19.42v 2467	. . . . 5 ⊢ (∃𝑦 ∈ 𝐷 (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜑)))
4	2, 3	bitri 173	. . . 4 ⊢ (∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜑)))
5	4	rexbii 2331	. . 3 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐶 (𝑥 = 𝐴 ∧ ∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜑)))
6		ceqsrex2v.1	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
7	6	anbi2d 437	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑦 = 𝐵 ∧ 𝜓)))
8	7	rexbidv 2327	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜓)))
9	8	ceqsrexv 2674	. . 3 ⊢ (𝐴 ∈ 𝐶 → (∃𝑥 ∈ 𝐶 (𝑥 = 𝐴 ∧ ∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜓)))
10	5, 9	syl5bb 181	. 2 ⊢ (𝐴 ∈ 𝐶 → (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜓)))
11		ceqsrex2v.2	. . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
12	11	ceqsrexv 2674	. 2 ⊢ (𝐵 ∈ 𝐷 → (∃𝑦 ∈ 𝐷 (𝑦 = 𝐵 ∧ 𝜓) ↔ 𝜒))
13	10, 12	sylan9bb 435	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∃𝑥 ∈ 𝐶 ∃𝑦 ∈ 𝐷 ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝜑) ↔ 𝜒))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∃wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559

This theorem is referenced by: (None)