Theorem raldifeq

Description: Equality theorem for restricted universal quantifier. (Contributed by Thierry Arnoux, 6-Jul-2019)

	Ref	Expression
Hypotheses	raldifeq.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	raldifeq.2	⊢ ( 𝜑 → ∀ 𝑥 ∈ ( 𝐵 ∖ 𝐴 ) 𝜓 )
Assertion	raldifeq	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) )

Proof

Step	Hyp	Ref	Expression
1		raldifeq.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		raldifeq.2	⊢ ( 𝜑 → ∀ 𝑥 ∈ ( 𝐵 ∖ 𝐴 ) 𝜓 )
3	2	biantrud	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ( ∀ 𝑥 ∈ 𝐴 𝜓 ∧ ∀ 𝑥 ∈ ( 𝐵 ∖ 𝐴 ) 𝜓 ) ) )
4		ralunb	⊢ ( ∀ 𝑥 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) 𝜓 ↔ ( ∀ 𝑥 ∈ 𝐴 𝜓 ∧ ∀ 𝑥 ∈ ( 𝐵 ∖ 𝐴 ) 𝜓 ) )
5	3 4	bitr4di	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) 𝜓 ) )
6		undif	⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = 𝐵 )
7	1 6	sylib	⊢ ( 𝜑 → ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) = 𝐵 )
8	7	raleqdv	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ ( 𝐴 ∪ ( 𝐵 ∖ 𝐴 ) ) 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) )
9	5 8	bitrd	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ∈ 𝐵 𝜓 ) )