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Metamath Proof Explorer

Theorem r19.29af

Description: A commonly used pattern based on r19.29 . See r19.29a , r19.29an for a variant when x is disjoint from ph . (Contributed by Thierry Arnoux, 29-Nov-2017)

	Ref	Expression
Hypotheses	r19.29af.0	⊢ Ⅎ 𝑥 𝜑
	r19.29af.1	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
	r19.29af.2	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
Assertion	r19.29af	⊢ ( 𝜑 → 𝜒 )

Proof

Step	Hyp	Ref	Expression
1		r19.29af.0	⊢ Ⅎ 𝑥 𝜑
2		r19.29af.1	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝜓 ) → 𝜒 )
3		r19.29af.2	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝜓 )
4		nfv	⊢ Ⅎ 𝑥 𝜒
5	1 4 2 3	r19.29af2	⊢ ( 𝜑 → 𝜒 )