Theorem rexdifi 4173

Description: Restricted existential quantification over a difference. (Contributed by AV, 25-Oct-2023.)

Assertion

Ref	Expression
rexdifi	⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)

Proof of Theorem rexdifi

Step	Hyp	Ref	Expression
1		df-rex 3077	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-ral 3068	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑))
3		nfa1 2152	. . . . 5 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑)
4		simprl 770	. . . . . . . 8 ⊢ ((∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 ∈ 𝐴)
5		con2 135	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝜑) → (𝜑 → ¬ 𝑥 ∈ 𝐵))
6	5	sps 2186	. . . . . . . . . . 11 ⊢ (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → (𝜑 → ¬ 𝑥 ∈ 𝐵))
7	6	com12 32	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → ¬ 𝑥 ∈ 𝐵))
8	7	adantl 481	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → ¬ 𝑥 ∈ 𝐵))
9	8	impcom 407	. . . . . . . 8 ⊢ ((∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) → ¬ 𝑥 ∈ 𝐵)
10	4, 9	eldifd 3987	. . . . . . 7 ⊢ ((∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 ∈ (𝐴 ∖ 𝐵))
11		simprr 772	. . . . . . 7 ⊢ ((∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝜑)
12	10, 11	jca 511	. . . . . 6 ⊢ ((∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) → (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑))
13	12	ex 412	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑)))
14	3, 13	eximd 2217	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑)))
15	14	impcom 407	. . 3 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑)) → ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑))
16	1, 2, 15	syl2anb 597	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝜑) → ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑))
17		df-rex 3077	. 2 ⊢ (∃𝑥 ∈ (𝐴 ∖ 𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑))
18	16, 17	sylibr 234	1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1535  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ∖ cdif 3973

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-10 2141  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-v 3490  df-dif 3979

This theorem is referenced by:  releldmdifi  8086