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Theorem rexdifi 4145
Description: Restricted existential quantification over a difference. (Contributed by AV, 25-Oct-2023.)
Assertion
Ref Expression
rexdifi ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴𝐵)𝜑)

Proof of Theorem rexdifi
StepHypRef Expression
1 df-rex 3070 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 df-ral 3061 . . 3 (∀𝑥𝐵 ¬ 𝜑 ↔ ∀𝑥(𝑥𝐵 → ¬ 𝜑))
3 nfa1 2147 . . . . 5 𝑥𝑥(𝑥𝐵 → ¬ 𝜑)
4 simprl 768 . . . . . . . 8 ((∀𝑥(𝑥𝐵 → ¬ 𝜑) ∧ (𝑥𝐴𝜑)) → 𝑥𝐴)
5 con2 135 . . . . . . . . . . . 12 ((𝑥𝐵 → ¬ 𝜑) → (𝜑 → ¬ 𝑥𝐵))
65sps 2177 . . . . . . . . . . 11 (∀𝑥(𝑥𝐵 → ¬ 𝜑) → (𝜑 → ¬ 𝑥𝐵))
76com12 32 . . . . . . . . . 10 (𝜑 → (∀𝑥(𝑥𝐵 → ¬ 𝜑) → ¬ 𝑥𝐵))
87adantl 481 . . . . . . . . 9 ((𝑥𝐴𝜑) → (∀𝑥(𝑥𝐵 → ¬ 𝜑) → ¬ 𝑥𝐵))
98impcom 407 . . . . . . . 8 ((∀𝑥(𝑥𝐵 → ¬ 𝜑) ∧ (𝑥𝐴𝜑)) → ¬ 𝑥𝐵)
104, 9eldifd 3959 . . . . . . 7 ((∀𝑥(𝑥𝐵 → ¬ 𝜑) ∧ (𝑥𝐴𝜑)) → 𝑥 ∈ (𝐴𝐵))
11 simprr 770 . . . . . . 7 ((∀𝑥(𝑥𝐵 → ¬ 𝜑) ∧ (𝑥𝐴𝜑)) → 𝜑)
1210, 11jca 511 . . . . . 6 ((∀𝑥(𝑥𝐵 → ¬ 𝜑) ∧ (𝑥𝐴𝜑)) → (𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
1312ex 412 . . . . 5 (∀𝑥(𝑥𝐵 → ¬ 𝜑) → ((𝑥𝐴𝜑) → (𝑥 ∈ (𝐴𝐵) ∧ 𝜑)))
143, 13eximd 2208 . . . 4 (∀𝑥(𝑥𝐵 → ¬ 𝜑) → (∃𝑥(𝑥𝐴𝜑) → ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑)))
1514impcom 407 . . 3 ((∃𝑥(𝑥𝐴𝜑) ∧ ∀𝑥(𝑥𝐵 → ¬ 𝜑)) → ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
161, 2, 15syl2anb 597 . 2 ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 ¬ 𝜑) → ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
17 df-rex 3070 . 2 (∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
1816, 17sylibr 233 1 ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴𝐵)𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1538  wex 1780  wcel 2105  wral 3060  wrex 3069  cdif 3945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-v 3475  df-dif 3951
This theorem is referenced by:  releldmdifi  8034
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