Theorem r19.29an 2612

Description: A commonly used pattern based on r19.29 2607. (Contributed by Thierry Arnoux, 29-Dec-2019.)

Hypothesis

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

Assertion

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

Distinct variable groups:   𝜒,𝑥   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem r19.29an

Step	Hyp	Ref	Expression
1		nfv 1521	. . 3 ⊢ Ⅎ𝑥𝜑
2		nfre1 2513	. . 3 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝜓
3	1, 2	nfan 1558	. 2 ⊢ Ⅎ𝑥(𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓)
4		r19.29an.1	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
5	4	adantllr 478	. 2 ⊢ ((((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
6		simpr 109	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 𝜓)
7	3, 5, 6	r19.29af 2611	1 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝜓) → 𝜒)

Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 2141  ∃wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-ral 2453  df-rex 2454

This theorem is referenced by:  exmidontriimlem2  7199  summodclem2  11345