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Theorem prsrcmpltd 34384
Description: If a statement is true for all pairs of elements of a class, all pairs of elements of its complement relative to a second class, and all pairs with one element in each, then it is true for all pairs of elements of the second class. (Contributed by BTernaryTau, 27-Sep-2023.)
Hypotheses
Ref Expression
prsrcmpltd.1 (𝜑 → ((𝐶𝐴𝐷𝐴) → 𝜓))
prsrcmpltd.2 (𝜑 → ((𝐶𝐴𝐷 ∈ (𝐵𝐴)) → 𝜓))
prsrcmpltd.3 (𝜑 → ((𝐶 ∈ (𝐵𝐴) ∧ 𝐷𝐴) → 𝜓))
prsrcmpltd.4 (𝜑 → ((𝐶 ∈ (𝐵𝐴) ∧ 𝐷 ∈ (𝐵𝐴)) → 𝜓))
Assertion
Ref Expression
prsrcmpltd (𝜑 → ((𝐶𝐵𝐷𝐵) → 𝜓))

Proof of Theorem prsrcmpltd
StepHypRef Expression
1 prsrcmpltd.1 . . . . . . 7 (𝜑 → ((𝐶𝐴𝐷𝐴) → 𝜓))
21expdimp 451 . . . . . 6 ((𝜑𝐶𝐴) → (𝐷𝐴𝜓))
3 prsrcmpltd.2 . . . . . . 7 (𝜑 → ((𝐶𝐴𝐷 ∈ (𝐵𝐴)) → 𝜓))
43expdimp 451 . . . . . 6 ((𝜑𝐶𝐴) → (𝐷 ∈ (𝐵𝐴) → 𝜓))
52, 4srcmpltd 34383 . . . . 5 ((𝜑𝐶𝐴) → (𝐷𝐵𝜓))
65impancom 450 . . . 4 ((𝜑𝐷𝐵) → (𝐶𝐴𝜓))
7 prsrcmpltd.3 . . . . . . 7 (𝜑 → ((𝐶 ∈ (𝐵𝐴) ∧ 𝐷𝐴) → 𝜓))
87expdimp 451 . . . . . 6 ((𝜑𝐶 ∈ (𝐵𝐴)) → (𝐷𝐴𝜓))
9 prsrcmpltd.4 . . . . . . 7 (𝜑 → ((𝐶 ∈ (𝐵𝐴) ∧ 𝐷 ∈ (𝐵𝐴)) → 𝜓))
109expdimp 451 . . . . . 6 ((𝜑𝐶 ∈ (𝐵𝐴)) → (𝐷 ∈ (𝐵𝐴) → 𝜓))
118, 10srcmpltd 34383 . . . . 5 ((𝜑𝐶 ∈ (𝐵𝐴)) → (𝐷𝐵𝜓))
1211impancom 450 . . . 4 ((𝜑𝐷𝐵) → (𝐶 ∈ (𝐵𝐴) → 𝜓))
136, 12srcmpltd 34383 . . 3 ((𝜑𝐷𝐵) → (𝐶𝐵𝜓))
1413ex 411 . 2 (𝜑 → (𝐷𝐵 → (𝐶𝐵𝜓)))
1514impcomd 410 1 (𝜑 → ((𝐶𝐵𝐷𝐵) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2104  cdif 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322
This theorem is referenced by:  f1resrcmplf1d  34388
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