Theorem prsrcmpltd 32626

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 456	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ 𝐴 → 𝜓))
3		prsrcmpltd.2	. . . . . . 7 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ (𝐵 ∖ 𝐴)) → 𝜓))
4	3	expdimp 456	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ (𝐵 ∖ 𝐴) → 𝜓))
5	2, 4	srcmpltd 32625	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐷 ∈ 𝐵 → 𝜓))
6	5	impancom 455	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∈ 𝐴 → 𝜓))
7		prsrcmpltd.3	. . . . . . 7 ⊢ (𝜑 → ((𝐶 ∈ (𝐵 ∖ 𝐴) ∧ 𝐷 ∈ 𝐴) → 𝜓))
8	7	expdimp 456	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵 ∖ 𝐴)) → (𝐷 ∈ 𝐴 → 𝜓))
9		prsrcmpltd.4	. . . . . . 7 ⊢ (𝜑 → ((𝐶 ∈ (𝐵 ∖ 𝐴) ∧ 𝐷 ∈ (𝐵 ∖ 𝐴)) → 𝜓))
10	9	expdimp 456	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵 ∖ 𝐴)) → (𝐷 ∈ (𝐵 ∖ 𝐴) → 𝜓))
11	8, 10	srcmpltd 32625	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵 ∖ 𝐴)) → (𝐷 ∈ 𝐵 → 𝜓))
12	11	impancom 455	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∈ (𝐵 ∖ 𝐴) → 𝜓))
13	6, 12	srcmpltd 32625	. . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ 𝐵) → (𝐶 ∈ 𝐵 → 𝜓))
14	13	ex 416	. 2 ⊢ (𝜑 → (𝐷 ∈ 𝐵 → (𝐶 ∈ 𝐵 → 𝜓)))
15	14	impcomd 415	1 ⊢ (𝜑 → ((𝐶 ∈ 𝐵 ∧ 𝐷 ∈ 𝐵) → 𝜓))

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2114   ∖ cdif 3840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212

This theorem is referenced by:  f1resrcmplf1d  32630