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Metamath Proof Explorer

Theorem subsubadd23

Description: Swap the second and the third terms in a difference of a difference and a sum. (Contributed by AV, 15-Nov-2025)

Ref Expression
Hypotheses negidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
pncand.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
subaddd.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
addsub4d.4 ⊢ ( 𝜑 → 𝐷 ∈ ℂ )
Assertion subsubadd23 ( 𝜑 → ( ( 𝐴 − 𝐵 ) − ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 − 𝐶 ) − ( 𝐵 + 𝐷 ) ) )

Proof

Step Hyp Ref Expression
1 negidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 pncand.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 subaddd.3 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
4 addsub4d.4 ⊢ ( 𝜑 → 𝐷 ∈ ℂ )
5 1 2 3 sub32d ⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) − 𝐶 ) = ( ( 𝐴 − 𝐶 ) − 𝐵 ) )
6 5 oveq1d ⊢ ( 𝜑 → ( ( ( 𝐴 − 𝐵 ) − 𝐶 ) − 𝐷 ) = ( ( ( 𝐴 − 𝐶 ) − 𝐵 ) − 𝐷 ) )
7 1 2 subcld ⊢ ( 𝜑 → ( 𝐴 − 𝐵 ) ∈ ℂ )
8 7 3 4 subsub4d ⊢ ( 𝜑 → ( ( ( 𝐴 − 𝐵 ) − 𝐶 ) − 𝐷 ) = ( ( 𝐴 − 𝐵 ) − ( 𝐶 + 𝐷 ) ) )
9 1 3 subcld ⊢ ( 𝜑 → ( 𝐴 − 𝐶 ) ∈ ℂ )
10 9 2 4 subsub4d ⊢ ( 𝜑 → ( ( ( 𝐴 − 𝐶 ) − 𝐵 ) − 𝐷 ) = ( ( 𝐴 − 𝐶 ) − ( 𝐵 + 𝐷 ) ) )
11 6 8 10 3eqtr3d ⊢ ( 𝜑 → ( ( 𝐴 − 𝐵 ) − ( 𝐶 + 𝐷 ) ) = ( ( 𝐴 − 𝐶 ) − ( 𝐵 + 𝐷 ) ) )