Theorem reusv3i 5422

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 2751	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑥 = 𝐷))
4	1, 3	imbi12d 344	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝜑 → 𝑥 = 𝐶) ↔ (𝜓 → 𝑥 = 𝐷)))
5	4	cbvralvw 3243	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷))
6	5	biimpi 216	. . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷))
7		raaanv 4541	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) ↔ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ∧ ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷)))
8		anim12 808	. . . . . 6 ⊢ (((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) → ((𝜑 ∧ 𝜓) → (𝑥 = 𝐶 ∧ 𝑥 = 𝐷)))
9		eqtr2 2764	. . . . . 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 3129	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
12	7, 11	sylbir 235	. . 3 ⊢ ((∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ∧ ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷)) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
13	6, 12	mpdan 686	. 2 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
14	13	rexlimivw 3157	1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537  ∀wral 3067  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-dif 3979  df-nul 4353

This theorem is referenced by:  reusv3  5423