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Metamath Proof Explorer

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 bitrid ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( 𝜓 ∨ 𝜒 ) ) )