Theorem rextpg

Description: Convert a restricted existential quantification over a triple to a disjunction. (Contributed by Mario Carneiro, 23-Apr-2015)

	Ref	Expression
Hypotheses	ralprg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	ralprg.2	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
	raltpg.3	⊢ ( 𝑥 = 𝐶 → ( 𝜑 ↔ 𝜃 ) )
Assertion	rextpg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 , 𝐶 } 𝜑 ↔ ( 𝜓 ∨ 𝜒 ∨ 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ralprg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		ralprg.2	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
3		raltpg.3	⊢ ( 𝑥 = 𝐶 → ( 𝜑 ↔ 𝜃 ) )
4	1 2	rexprg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( 𝜓 ∨ 𝜒 ) ) )
5	4	orbi1d	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ∨ ∃ 𝑥 ∈ { 𝐶 } 𝜑 ) ↔ ( ( 𝜓 ∨ 𝜒 ) ∨ ∃ 𝑥 ∈ { 𝐶 } 𝜑 ) ) )
6	3	rexsng	⊢ ( 𝐶 ∈ 𝑋 → ( ∃ 𝑥 ∈ { 𝐶 } 𝜑 ↔ 𝜃 ) )
7	6	orbi2d	⊢ ( 𝐶 ∈ 𝑋 → ( ( ( 𝜓 ∨ 𝜒 ) ∨ ∃ 𝑥 ∈ { 𝐶 } 𝜑 ) ↔ ( ( 𝜓 ∨ 𝜒 ) ∨ 𝜃 ) ) )
8	5 7	sylan9bb	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ 𝐶 ∈ 𝑋 ) → ( ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ∨ ∃ 𝑥 ∈ { 𝐶 } 𝜑 ) ↔ ( ( 𝜓 ∨ 𝜒 ) ∨ 𝜃 ) ) )
9	8	3impa	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ∨ ∃ 𝑥 ∈ { 𝐶 } 𝜑 ) ↔ ( ( 𝜓 ∨ 𝜒 ) ∨ 𝜃 ) ) )
10		df-tp	⊢ { 𝐴 , 𝐵 , 𝐶 } = ( { 𝐴 , 𝐵 } ∪ { 𝐶 } )
11	10	rexeqi	⊢ ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 , 𝐶 } 𝜑 ↔ ∃ 𝑥 ∈ ( { 𝐴 , 𝐵 } ∪ { 𝐶 } ) 𝜑 )
12		rexun	⊢ ( ∃ 𝑥 ∈ ( { 𝐴 , 𝐵 } ∪ { 𝐶 } ) 𝜑 ↔ ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ∨ ∃ 𝑥 ∈ { 𝐶 } 𝜑 ) )
13	11 12	bitri	⊢ ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 , 𝐶 } 𝜑 ↔ ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ∨ ∃ 𝑥 ∈ { 𝐶 } 𝜑 ) )
14		df-3or	⊢ ( ( 𝜓 ∨ 𝜒 ∨ 𝜃 ) ↔ ( ( 𝜓 ∨ 𝜒 ) ∨ 𝜃 ) )
15	9 13 14	3bitr4g	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ) → ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 , 𝐶 } 𝜑 ↔ ( 𝜓 ∨ 𝜒 ∨ 𝜃 ) ) )