Theorem r19.29a 2637

Description: A commonly used pattern based on r19.29 2631. (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 1539	. 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 2635	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2164  ∃wrex 2473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-i5r 1546

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-ral 2477  df-rex 2478

This theorem is referenced by:  cnegexlem3  8196  cnegex  8197  modqmuladdnn0  10439  uzwodc  12174  1arith  12505  mhmid  13185  mhmmnd  13186  ghmgrp  13188  ghmcmn  13397  ringinvnz1ne0  13545  neitx  14436