Theorem r19.29a 2613

Description: A commonly used pattern based on r19.29 2607. (Contributed by Thierry Arnoux, 22-Nov-2017.)

Hypotheses

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

Assertion

Ref	Expression
r19.29a	⊢ (𝜑 → 𝜒)

Distinct variable groups:   𝜒,𝑥   𝜑,𝑥

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

Proof of Theorem r19.29a

Step	Hyp	Ref	Expression
1		nfv 1521	. 2 ⊢ Ⅎ𝑥𝜑
2		r19.29a.1	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
3		r19.29a.2	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
4	1, 2, 3	r19.29af 2611	1 ⊢ (𝜑 → 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:  cnegexlem3  8096  cnegex  8097  modqmuladdnn0  10324  uzwodc  11992  1arith  12319  neitx  13062