Theorem reusv3i 5402

Description: Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)

Hypotheses

Ref	Expression
reusv3.1	⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
reusv3.2	⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷)

Assertion

Ref	Expression
reusv3i	⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))

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

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

Proof of Theorem reusv3i

Step	Hyp	Ref	Expression
1		reusv3.1	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓))
2		reusv3.2	. . . . . . 7 ⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷)
3	2	eqeq2d 2742	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑥 = 𝐷))
4	1, 3	imbi12d 344	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝜑 → 𝑥 = 𝐶) ↔ (𝜓 → 𝑥 = 𝐷)))
5	4	cbvralvw 3233	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷))
6	5	biimpi 215	. . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷))
7		raaanv 4521	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) ↔ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ∧ ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷)))
8		anim12 806	. . . . . 6 ⊢ (((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) → ((𝜑 ∧ 𝜓) → (𝑥 = 𝐶 ∧ 𝑥 = 𝐷)))
9		eqtr2 2755	. . . . . 6 ⊢ ((𝑥 = 𝐶 ∧ 𝑥 = 𝐷) → 𝐶 = 𝐷)
10	8, 9	syl6 35	. . . . 5 ⊢ (((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) → ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
11	10	2ralimi 3122	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
12	7, 11	sylbir 234	. . 3 ⊢ ((∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ∧ ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷)) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
13	6, 12	mpdan 684	. 2 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
14	13	rexlimivw 3150	1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540  ∀wral 3060  ∃wrex 3069

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-dif 3951  df-nul 4323

This theorem is referenced by:  reusv3  5403