Theorem psseq1

Description: Equality theorem for proper subclass. (Contributed by NM, 7-Feb-1996)

	Ref	Expression
Assertion	psseq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		sseq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶 ) )
2		neeq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶 ) )
3	1 2	anbi12d	⊢ ( 𝐴 = 𝐵 → ( ( 𝐴 ⊆ 𝐶 ∧ 𝐴 ≠ 𝐶 ) ↔ ( 𝐵 ⊆ 𝐶 ∧ 𝐵 ≠ 𝐶 ) ) )
4		df-pss	⊢ ( 𝐴 ⊊ 𝐶 ↔ ( 𝐴 ⊆ 𝐶 ∧ 𝐴 ≠ 𝐶 ) )
5		df-pss	⊢ ( 𝐵 ⊊ 𝐶 ↔ ( 𝐵 ⊆ 𝐶 ∧ 𝐵 ≠ 𝐶 ) )
6	3 4 5	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶 ) )