Theorem r19.29an

Description: A commonly used pattern in the spirit of r19.29 . (Contributed by Thierry Arnoux, 29-Dec-2019) (Proof shortened by Wolf Lammen, 17-Jun-2023)

	Ref	Expression
Hypothesis	rexlimdva2.1	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
Assertion	r19.29an	⊢ ( ( 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 𝜓 ) → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		rexlimdva2.1	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
2	1	rexlimdva2	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜒 ) )
3	2	imp	⊢ ( ( 𝜑 ∧ ∃ 𝑥 ∈ 𝐴 𝜓 ) → 𝜒 )