Theorem r19.29af2

Description: A commonly used pattern based on r19.29 . (Contributed by Thierry Arnoux, 17-Dec-2017) (Proof shortened by OpenAI, 25-Mar-2020)

	Ref	Expression
Hypotheses	r19.29af2.p	⊢ Ⅎ 𝑥 𝜑
	r19.29af2.c	⊢ Ⅎ 𝑥 𝜒
	r19.29af2.1	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
	r19.29af2.2	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
Assertion	r19.29af2	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		r19.29af2.p	⊢ Ⅎ 𝑥 𝜑
2		r19.29af2.c	⊢ Ⅎ 𝑥 𝜒
3		r19.29af2.1	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
4		r19.29af2.2	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
5	3	exp31	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝜓 → 𝜒 ) ) )
6	1 2 5	rexlimd	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝜓 → 𝜒 ) )
7	4 6	mpd	⊢ ( 𝜑 → 𝜒 )