Theorem bj-2upleq

Description: Substitution property for (| - ,, - |) . (Contributed by BJ, 6-Oct-2018)

	Ref	Expression
Assertion	bj-2upleq	⊢ ( 𝐴 = 𝐵 → ( 𝐶 = 𝐷 → ⦅ 𝐴 , 𝐶 ⦆ = ⦅ 𝐵 , 𝐷 ⦆ ) )

Proof

Step	Hyp	Ref	Expression
1		bj-1upleq	⊢ ( 𝐴 = 𝐵 → ⦅ 𝐴 ⦆ = ⦅ 𝐵 ⦆ )
2		bj-xtageq	⊢ ( 𝐶 = 𝐷 → ( { 1o } × tag 𝐶 ) = ( { 1o } × tag 𝐷 ) )
3		uneq12	⊢ ( ( ⦅ 𝐴 ⦆ = ⦅ 𝐵 ⦆ ∧ ( { 1o } × tag 𝐶 ) = ( { 1o } × tag 𝐷 ) ) → ( ⦅ 𝐴 ⦆ ∪ ( { 1o } × tag 𝐶 ) ) = ( ⦅ 𝐵 ⦆ ∪ ( { 1o } × tag 𝐷 ) ) )
4	3	ex	⊢ ( ⦅ 𝐴 ⦆ = ⦅ 𝐵 ⦆ → ( ( { 1o } × tag 𝐶 ) = ( { 1o } × tag 𝐷 ) → ( ⦅ 𝐴 ⦆ ∪ ( { 1o } × tag 𝐶 ) ) = ( ⦅ 𝐵 ⦆ ∪ ( { 1o } × tag 𝐷 ) ) ) )
5	1 2 4	syl2im	⊢ ( 𝐴 = 𝐵 → ( 𝐶 = 𝐷 → ( ⦅ 𝐴 ⦆ ∪ ( { 1o } × tag 𝐶 ) ) = ( ⦅ 𝐵 ⦆ ∪ ( { 1o } × tag 𝐷 ) ) ) )
6		df-bj-2upl	⊢ ⦅ 𝐴 , 𝐶 ⦆ = ( ⦅ 𝐴 ⦆ ∪ ( { 1o } × tag 𝐶 ) )
7		df-bj-2upl	⊢ ⦅ 𝐵 , 𝐷 ⦆ = ( ⦅ 𝐵 ⦆ ∪ ( { 1o } × tag 𝐷 ) )
8	6 7	eqeq12i	⊢ ( ⦅ 𝐴 , 𝐶 ⦆ = ⦅ 𝐵 , 𝐷 ⦆ ↔ ( ⦅ 𝐴 ⦆ ∪ ( { 1o } × tag 𝐶 ) ) = ( ⦅ 𝐵 ⦆ ∪ ( { 1o } × tag 𝐷 ) ) )
9	5 8	syl6ibr	⊢ ( 𝐴 = 𝐵 → ( 𝐶 = 𝐷 → ⦅ 𝐴 , 𝐶 ⦆ = ⦅ 𝐵 , 𝐷 ⦆ ) )