Theorem r19.29a

Description: A commonly used pattern in the spirit of r19.29 . (Contributed by Thierry Arnoux, 22-Nov-2017) Reduce axiom usage. (Revised by Wolf Lammen, 17-Jun-2023)

	Ref	Expression
Hypotheses	r19.29a.1	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
	r19.29a.2	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
Assertion	r19.29a	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		r19.29a.1	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
2		r19.29a.2	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
3	1	rexlimdva2	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜒 ) )
4	2 3	mpd	⊢ ( 𝜑 → 𝜒 )