Theorem rexprg

Description: Convert a restricted existential quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011) (Revised by Mario Carneiro, 23-Apr-2015) Avoid ax-10 , ax-12 . (Revised by Gino Giotto, 30-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		ralprg.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2		ralprg.2	⊢ ( 𝑥 = 𝐵 → ( 𝜑 ↔ 𝜒 ) )
3	1	notbid	⊢ ( 𝑥 = 𝐴 → ( ¬ 𝜑 ↔ ¬ 𝜓 ) )
4	2	notbid	⊢ ( 𝑥 = 𝐵 → ( ¬ 𝜑 ↔ ¬ 𝜒 ) )
5	3 4	ralprg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } ¬ 𝜑 ↔ ( ¬ 𝜓 ∧ ¬ 𝜒 ) ) )
6		ralnex	⊢ ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } ¬ 𝜑 ↔ ¬ ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 )
7		pm4.56	⊢ ( ( ¬ 𝜓 ∧ ¬ 𝜒 ) ↔ ¬ ( 𝜓 ∨ 𝜒 ) )
8	6 7	bibi12i	⊢ ( ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } ¬ 𝜑 ↔ ( ¬ 𝜓 ∧ ¬ 𝜒 ) ) ↔ ( ¬ ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ¬ ( 𝜓 ∨ 𝜒 ) ) )
9		notbi	⊢ ( ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( 𝜓 ∨ 𝜒 ) ) ↔ ( ¬ ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ¬ ( 𝜓 ∨ 𝜒 ) ) )
10	8 9	sylbb2	⊢ ( ( ∀ 𝑥 ∈ { 𝐴 , 𝐵 } ¬ 𝜑 ↔ ( ¬ 𝜓 ∧ ¬ 𝜒 ) ) → ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( 𝜓 ∨ 𝜒 ) ) )
11	5 10	syl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑥 ∈ { 𝐴 , 𝐵 } 𝜑 ↔ ( 𝜓 ∨ 𝜒 ) ) )