Theorem srcmpltd 32456

Description: If a statement is true for every element of a class and for every element of its complement relative to a second class, then it is true for every element in the second class. (Contributed by BTernaryTau, 27-Sep-2023.)

Hypotheses

Ref	Expression
srcmpltd.1	⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝜓))
srcmpltd.2	⊢ (𝜑 → (𝐶 ∈ (𝐵 ∖ 𝐴) → 𝜓))

Assertion

Ref	Expression
srcmpltd	⊢ (𝜑 → (𝐶 ∈ 𝐵 → 𝜓))

Proof of Theorem srcmpltd

Step	Hyp	Ref	Expression
1		elun2 4104	. . 3 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ (𝐴 ∪ 𝐵))
2		undif2 4383	. . 3 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)
3	1, 2	eleqtrrdi 2901	. 2 ⊢ (𝐶 ∈ 𝐵 → 𝐶 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴)))
4		srcmpltd.1	. . 3 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝜓))
5		srcmpltd.2	. . 3 ⊢ (𝜑 → (𝐶 ∈ (𝐵 ∖ 𝐴) → 𝜓))
6		elunant 4105	. . 3 ⊢ ((𝐶 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴)) → 𝜓) ↔ ((𝐶 ∈ 𝐴 → 𝜓) ∧ (𝐶 ∈ (𝐵 ∖ 𝐴) → 𝜓)))
7	4, 5, 6	sylanbrc 586	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴 ∪ (𝐵 ∖ 𝐴)) → 𝜓))
8	3, 7	syl5 34	1 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → 𝜓))

Syntax hints:   → wi 4   ∈ wcel 2111   ∖ cdif 3878   ∪ cun 3879

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244

This theorem is referenced by:  prsrcmpltd  32457