Theorem ceqsexv2d

Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016)

	Ref	Expression
Hypotheses	ceqsexv2d.1	⊢ 𝐴 ∈ V
	ceqsexv2d.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	ceqsexv2d.3	⊢ 𝜓
Assertion	ceqsexv2d	⊢ ∃ 𝑥 𝜑

Proof

Step	Hyp	Ref	Expression
1		ceqsexv2d.1	⊢ 𝐴 ∈ V
2		ceqsexv2d.2	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
3		ceqsexv2d.3	⊢ 𝜓
4	1 2	ceqsexv	⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) ↔ 𝜓 )
5	4	biimpri	⊢ ( 𝜓 → ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) )
6		exsimpr	⊢ ( ∃ 𝑥 ( 𝑥 = 𝐴 ∧ 𝜑 ) → ∃ 𝑥 𝜑 )
7	3 5 6	mp2b	⊢ ∃ 𝑥 𝜑