Theorem psseq2

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

	Ref	Expression
Assertion	psseq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sseq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ⊆ 𝐴 ↔ 𝐶 ⊆ 𝐵 ) )
2		neeq2	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ≠ 𝐴 ↔ 𝐶 ≠ 𝐵 ) )
3	1 2	anbi12d	⊢ ( 𝐴 = 𝐵 → ( ( 𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ 𝐴 ) ↔ ( 𝐶 ⊆ 𝐵 ∧ 𝐶 ≠ 𝐵 ) ) )
4		df-pss	⊢ ( 𝐶 ⊊ 𝐴 ↔ ( 𝐶 ⊆ 𝐴 ∧ 𝐶 ≠ 𝐴 ) )
5		df-pss	⊢ ( 𝐶 ⊊ 𝐵 ↔ ( 𝐶 ⊆ 𝐵 ∧ 𝐶 ≠ 𝐵 ) )
6	3 4 5	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵 ) )