Theorem reusv3i 4380

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 2151	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑥 = 𝐷))
4	1, 3	imbi12d 233	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝜑 → 𝑥 = 𝐶) ↔ (𝜓 → 𝑥 = 𝐷)))
5	4	cbvralv 2654	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷))
6	5	biimpi 119	. . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷))
7		raaanv 3470	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) ↔ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ∧ ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷)))
8		anim12 341	. . . . . . 7 ⊢ (((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) → ((𝜑 ∧ 𝜓) → (𝑥 = 𝐶 ∧ 𝑥 = 𝐷)))
9		eqtr2 2158	. . . . . . 7 ⊢ ((𝑥 = 𝐶 ∧ 𝑥 = 𝐷) → 𝐶 = 𝐷)
10	8, 9	syl6 33	. . . . . 6 ⊢ (((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) → ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
11	10	ralimi 2495	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) → ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
12	11	ralimi 2495	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
13	7, 12	sylbir 134	. . 3 ⊢ ((∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ∧ ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷)) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
14	6, 13	mpdan 417	. 2 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
15	14	rexlimivw 2545	1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  ∀wral 2416  ∃wrex 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422

This theorem is referenced by:  reusv3  4381