Theorem reusv1OLD 4867

Description: Obsolete proof of reusv1 4866 as of 7-Aug-2021. (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
reusv1OLD	⊢ (∃𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦)   𝐶(𝑦)

Proof of Theorem reusv1OLD

Step	Hyp	Ref	Expression
1		nfra1 2941	. . . 4 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
2	1	nfmo 2487	. . 3 ⊢ Ⅎ𝑦∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)
3		rsp 2929	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → (𝑦 ∈ 𝐵 → (𝜑 → 𝑥 = 𝐶)))
4	3	impd 447	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ((𝑦 ∈ 𝐵 ∧ 𝜑) → 𝑥 = 𝐶))
5	4	com12 32	. . . . . 6 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → 𝑥 = 𝐶))
6	5	alrimiv 1855	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∀𝑥(∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → 𝑥 = 𝐶))
7		moeq 3382	. . . . 5 ⊢ ∃*𝑥 𝑥 = 𝐶
8		moim 2519	. . . . 5 ⊢ (∀𝑥(∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → 𝑥 = 𝐶) → (∃*𝑥 𝑥 = 𝐶 → ∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
9	6, 7, 8	mpisyl 21	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜑) → ∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
10	9	ex 450	. . 3 ⊢ (𝑦 ∈ 𝐵 → (𝜑 → ∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
11	2, 10	rexlimi 3024	. 2 ⊢ (∃𝑦 ∈ 𝐵 𝜑 → ∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
12		mormo 3158	. 2 ⊢ (∃*𝑥∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))
13		reu5 3159	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ∧ ∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
14	13	rbaib 947	. 2 ⊢ (∃*𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))
15	11, 12, 14	3syl 18	1 ⊢ (∃𝑦 ∈ 𝐵 𝜑 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∃*wmo 2471  ∀wral 2912  ∃wrex 2913  ∃!wreu 2914  ∃*wrmo 2915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-v 3202

This theorem is referenced by: (None)