Theorem prsrcmpltd 32347

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 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → 𝜓))
prsrcmpltd.2	⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ (𝐵 ∖ 𝐴)) → 𝜓))
prsrcmpltd.3	⊢ (𝜑 → ((𝐶 ∈ (𝐵 ∖ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝜓))
prsrcmpltd.4	⊢ (𝜑 → ((𝐶 ∈ (𝐵 ∖ 𝐴) ∧ 𝐷 ∈ (𝐵 ∖ 𝐴)) → 𝜓))

Assertion

Ref	Expression
prsrcmpltd	⊢ (𝜑 → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → 𝜓))

Proof of Theorem prsrcmpltd

Step	Hyp	Ref	Expression
1		prsrcmpltd.1	. . . . . . 7 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → 𝜓))
2	1	expdimp 455	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ 𝐴 → 𝜓))
3		prsrcmpltd.2	. . . . . . 7 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ (𝐵 ∖ 𝐴)) → 𝜓))
4	3	expdimp 455	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ (𝐵 ∖ 𝐴) → 𝜓))
5	2, 4	srcmpltd 32346	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ 𝐵 → 𝜓))
6	5	impancom 454	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∈ 𝐴 → 𝜓))
7		prsrcmpltd.3	. . . . . . 7 ⊢ (𝜑 → ((𝐶 ∈ (𝐵 ∖ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝜓))
8	7	expdimp 455	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵 ∖ 𝐴)) → (𝐷 ∈ 𝐴 → 𝜓))
9		prsrcmpltd.4	. . . . . . 7 ⊢ (𝜑 → ((𝐶 ∈ (𝐵 ∖ 𝐴) ∧ 𝐷 ∈ (𝐵 ∖ 𝐴)) → 𝜓))
10	9	expdimp 455	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵 ∖ 𝐴)) → (𝐷 ∈ (𝐵 ∖ 𝐴) → 𝜓))
11	8, 10	srcmpltd 32346	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵 ∖ 𝐴)) → (𝐷 ∈ 𝐵 → 𝜓))
12	11	impancom 454	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∈ (𝐵 ∖ 𝐴) → 𝜓))
13	6, 12	srcmpltd 32346	. . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∈ 𝐵 → 𝜓))
14	13	ex 415	. 2 ⊢ (𝜑 → (𝐷 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝜓)))
15	14	impcomd 414	1 ⊢ (𝜑 → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → 𝜓))

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114   ∖ cdif 3933

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292

This theorem is referenced by:  f1resrcmplf1d  32360