Theorem rexdifi 4146

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 3072	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-ral 3063	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑))
3		nfa1 2149	. . . . 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 2179	. . . . . . . . . . 11 ⊢ (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → (𝜑 → ¬ 𝑥 ∈ 𝐵))
7	6	com12 32	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → ¬ 𝑥 ∈ 𝐵))
8	7	adantl 483	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → ¬ 𝑥 ∈ 𝐵))
9	8	impcom 409	. . . . . . . 8 ⊢ ((∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) → ¬ 𝑥 ∈ 𝐵)
10	4, 9	eldifd 3960	. . . . . . 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 513	. . . . . 6 ⊢ ((∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) → (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑))
13	12	ex 414	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑)))
14	3, 13	eximd 2210	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑)))
15	14	impcom 409	. . 3 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑)) → ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑))
16	1, 2, 15	syl2anb 599	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝜑) → ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑))
17		df-rex 3072	. 2 ⊢ (∃𝑥 ∈ (𝐴 ∖ 𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑))
18	16, 17	sylibr 233	1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397  ∀wal 1540  ∃wex 1782   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   ∖ cdif 3946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3952

This theorem is referenced by:  releldmdifi  8031