Theorem rexdifi 4103

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 3087	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		df-ral 3077	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑))
3		nfa1 2185	. . . . 5 ⊢ Ⅎ𝑥∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑)
4		simprl 780	. . . . . . . 8 ⊢ ((∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 ∈ 𝐴)
5		con2 135	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝜑) → (𝜑 → ¬ 𝑥 ∈ 𝐵))
6	5	sps 2220	. . . . . . . . . . 11 ⊢ (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → (𝜑 → ¬ 𝑥 ∈ 𝐵))
7	6	com12 32	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → ¬ 𝑥 ∈ 𝐵))
8	7	adantl 485	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → ¬ 𝑥 ∈ 𝐵))
9	8	impcom 411	. . . . . . . 8 ⊢ ((∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) → ¬ 𝑥 ∈ 𝐵)
10	4, 9	eldifd 3915	. . . . . . 7 ⊢ ((∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝑥 ∈ (𝐴 ∖ 𝐵))
11		simprr 782	. . . . . . 7 ⊢ ((∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) → 𝜑)
12	10, 11	jca 519	. . . . . 6 ⊢ ((∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) → (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑))
13	12	ex 416	. . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑)))
14	3, 13	eximd 2251	. . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑) → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑)))
15	14	impcom 411	. . 3 ⊢ ((∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ∧ ∀𝑥(𝑥 ∈ 𝐵 → ¬ 𝜑)) → ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑))
16	1, 2, 15	syl2anb 607	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝜑) → ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑))
17		df-rex 3087	. 2 ⊢ (∃𝑥 ∈ (𝐴 ∖ 𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐴 ∖ 𝐵) ∧ 𝜑))
18	16, 17	sylibr 236	1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1558  ∃wex 1799   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086   ∖ cdif 3901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-v 3456  df-dif 3907

This theorem is referenced by:  releldmdifi  8026