Theorem eqvincg

Description: A variable introduction law for class equality, closed form. (Contributed by Thierry Arnoux, 2-Mar-2017)

	Ref	Expression
Assertion	eqvincg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 = 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elisset	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
2		ax-1	⊢ ( 𝑥 = 𝐴 → ( 𝐴 = 𝐵 → 𝑥 = 𝐴 ) )
3		eqtr	⊢ ( ( 𝑥 = 𝐴 ∧ 𝐴 = 𝐵 ) → 𝑥 = 𝐵 )
4	3	ex	⊢ ( 𝑥 = 𝐴 → ( 𝐴 = 𝐵 → 𝑥 = 𝐵 ) )
5	2 4	jca	⊢ ( 𝑥 = 𝐴 → ( ( 𝐴 = 𝐵 → 𝑥 = 𝐴 ) ∧ ( 𝐴 = 𝐵 → 𝑥 = 𝐵 ) ) )
6	5	eximi	⊢ ( ∃ 𝑥 𝑥 = 𝐴 → ∃ 𝑥 ( ( 𝐴 = 𝐵 → 𝑥 = 𝐴 ) ∧ ( 𝐴 = 𝐵 → 𝑥 = 𝐵 ) ) )
7		pm3.43	⊢ ( ( ( 𝐴 = 𝐵 → 𝑥 = 𝐴 ) ∧ ( 𝐴 = 𝐵 → 𝑥 = 𝐵 ) ) → ( 𝐴 = 𝐵 → ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) ) )
8	7	eximi	⊢ ( ∃ 𝑥 ( ( 𝐴 = 𝐵 → 𝑥 = 𝐴 ) ∧ ( 𝐴 = 𝐵 → 𝑥 = 𝐵 ) ) → ∃ 𝑥 ( 𝐴 = 𝐵 → ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) ) )
9	1 6 8	3syl	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 ( 𝐴 = 𝐵 → ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) ) )
10		19.37v	⊢ ( ∃ 𝑥 ( 𝐴 = 𝐵 → ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) ) ↔ ( 𝐴 = 𝐵 → ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) ) )
11	9 10	sylib	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 = 𝐵 → ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) ) )
12		eqtr2	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) → 𝐴 = 𝐵 )
13	12	exlimiv	⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) → 𝐴 = 𝐵 )
14	11 13	impbid1	⊢ ( 𝐴 ∈ 𝑉 → ( 𝐴 = 𝐵 ↔ ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝑥 = 𝐵 ) ) )