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Theorem rexdifi 4108
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 3139 . . 3 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
2 df-ral 3138 . . 3 (∀𝑥𝐵 ¬ 𝜑 ↔ ∀𝑥(𝑥𝐵 → ¬ 𝜑))
3 nfa1 2156 . . . . 5 𝑥𝑥(𝑥𝐵 → ¬ 𝜑)
4 simprl 770 . . . . . . . 8 ((∀𝑥(𝑥𝐵 → ¬ 𝜑) ∧ (𝑥𝐴𝜑)) → 𝑥𝐴)
5 con2 137 . . . . . . . . . . . 12 ((𝑥𝐵 → ¬ 𝜑) → (𝜑 → ¬ 𝑥𝐵))
65sps 2186 . . . . . . . . . . 11 (∀𝑥(𝑥𝐵 → ¬ 𝜑) → (𝜑 → ¬ 𝑥𝐵))
76com12 32 . . . . . . . . . 10 (𝜑 → (∀𝑥(𝑥𝐵 → ¬ 𝜑) → ¬ 𝑥𝐵))
87adantl 485 . . . . . . . . 9 ((𝑥𝐴𝜑) → (∀𝑥(𝑥𝐵 → ¬ 𝜑) → ¬ 𝑥𝐵))
98impcom 411 . . . . . . . 8 ((∀𝑥(𝑥𝐵 → ¬ 𝜑) ∧ (𝑥𝐴𝜑)) → ¬ 𝑥𝐵)
104, 9eldifd 3930 . . . . . . 7 ((∀𝑥(𝑥𝐵 → ¬ 𝜑) ∧ (𝑥𝐴𝜑)) → 𝑥 ∈ (𝐴𝐵))
11 simprr 772 . . . . . . 7 ((∀𝑥(𝑥𝐵 → ¬ 𝜑) ∧ (𝑥𝐴𝜑)) → 𝜑)
1210, 11jca 515 . . . . . 6 ((∀𝑥(𝑥𝐵 → ¬ 𝜑) ∧ (𝑥𝐴𝜑)) → (𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
1312ex 416 . . . . 5 (∀𝑥(𝑥𝐵 → ¬ 𝜑) → ((𝑥𝐴𝜑) → (𝑥 ∈ (𝐴𝐵) ∧ 𝜑)))
143, 13eximd 2218 . . . 4 (∀𝑥(𝑥𝐵 → ¬ 𝜑) → (∃𝑥(𝑥𝐴𝜑) → ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑)))
1514impcom 411 . . 3 ((∃𝑥(𝑥𝐴𝜑) ∧ ∀𝑥(𝑥𝐵 → ¬ 𝜑)) → ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
161, 2, 15syl2anb 600 . 2 ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 ¬ 𝜑) → ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
17 df-rex 3139 . 2 (∃𝑥 ∈ (𝐴𝐵)𝜑 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐵) ∧ 𝜑))
1816, 17sylibr 237 1 ((∃𝑥𝐴 𝜑 ∧ ∀𝑥𝐵 ¬ 𝜑) → ∃𝑥 ∈ (𝐴𝐵)𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wal 1536  wex 1781  wcel 2115  wral 3133  wrex 3134  cdif 3916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922
This theorem is referenced by:  releldmdifi  7739
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