Theorem reusv3i 4494

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 2208	. . . . . 6 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑥 = 𝐷))
4	1, 3	imbi12d 234	. . . . 5 ⊢ (𝑦 = 𝑧 → ((𝜑 → 𝑥 = 𝐶) ↔ (𝜓 → 𝑥 = 𝐷)))
5	4	cbvralv 2729	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷))
6	5	biimpi 120	. . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷))
7		raaanv 3557	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) ↔ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ∧ ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷)))
8		anim12 344	. . . . . . 7 ⊢ (((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) → ((𝜑 ∧ 𝜓) → (𝑥 = 𝐶 ∧ 𝑥 = 𝐷)))
9		eqtr2 2215	. . . . . . 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 2560	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) → ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
12	11	ralimi 2560	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 → 𝑥 = 𝐶) ∧ (𝜓 → 𝑥 = 𝐷)) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
13	7, 12	sylbir 135	. . 3 ⊢ ((∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ∧ ∀𝑧 ∈ 𝐵 (𝜓 → 𝑥 = 𝐷)) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
14	6, 13	mpdan 421	. 2 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))
15	14	rexlimivw 2610	1 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∀wral 2475  ∃wrex 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481

This theorem is referenced by:  reusv3  4495