Theorem rexprgf

Description: Convert a restricted existential quantification over a pair to a disjunction, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 17-Sep-2011) (Revised by AV, 2-Apr-2023)

	Ref	Expression
Hypotheses	ralprgf.1	⊢ Ⅎ 𝑥 𝜓
	ralprgf.2	⊢ Ⅎ 𝑥 𝜒
	ralprgf.a	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	ralprgf.b	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
Assertion	rexprgf	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( 𝜓 ∨ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ralprgf.1	⊢ Ⅎ 𝑥 𝜓
2		ralprgf.2	⊢ Ⅎ 𝑥 𝜒
3		ralprgf.a	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
4		ralprgf.b	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
5		df-pr	⊢ { 𝐴 , 𝐵 } = ( { 𝐴 } ∪ { 𝐵 } )
6	5	rexeqi	⊢ ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ∃ 𝑥 ∈ ( { 𝐴 } ∪ { 𝐵 } ) 𝜑 )
7		rexun	⊢ ( ∃ 𝑥 ∈ ( { 𝐴 } ∪ { 𝐵 } ) 𝜑 ↔ ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ∨ ∃ 𝑥 ∈ { 𝐵 } 𝜑 ) )
8	6 7	bitri	⊢ ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ∨ ∃ 𝑥 ∈ { 𝐵 } 𝜑 ) )
9	1 3	rexsngf	⊢ ( 𝐴 ∈ 𝑉 → ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ↔ 𝜓 ) )
10	9	orbi1d	⊢ ( 𝐴 ∈ 𝑉 → ( ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ∨ ∃ 𝑥 ∈ { 𝐵 } 𝜑 ) ↔ ( 𝜓 ∨ ∃ 𝑥 ∈ { 𝐵 } 𝜑 ) ) )
11	2 4	rexsngf	⊢ ( 𝐵 ∈ 𝑊 → ( ∃ 𝑥 ∈ { 𝐵 } 𝜑 ↔ 𝜒 ) )
12	11	orbi2d	⊢ ( 𝐵 ∈ 𝑊 → ( ( 𝜓 ∨ ∃ 𝑥 ∈ { 𝐵 } 𝜑 ) ↔ ( 𝜓 ∨ 𝜒 ) ) )
13	10 12	sylan9bb	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( ∃ 𝑥 ∈ { 𝐴 } 𝜑 ∨ ∃ 𝑥 ∈ { 𝐵 } 𝜑 ) ↔ ( 𝜓 ∨ 𝜒 ) ) )
14	8 13	syl5bb	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( 𝜓 ∨ 𝜒 ) ) )