Theorem r19.29af2 3271

Description: A commonly used pattern based on r19.29 3126. (Contributed by Thierry Arnoux, 17-Dec-2017.) (Proof shortened by OpenAI, 25-Mar-2020.)

Hypotheses

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

Assertion

Ref	Expression
r19.29af2	⊢ (𝜑 → 𝜒)

Proof of Theorem r19.29af2

Step	Hyp	Ref	Expression
1		r19.29af2.2	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
2		r19.29af2.p	. . 3 ⊢ Ⅎ𝑥𝜑
3		r19.29af2.c	. . 3 ⊢ Ⅎ𝑥𝜒
4		r19.29af2.1	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
5	4	exp31 425	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
6	2, 3, 5	rexlimd 3270	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
7	1, 6	mpd 16	1 ⊢ (𝜑 → 𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  Ⅎwnf 1816   ∈ wcel 2145  ∃wrex 3087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817  df-ral 3078  df-rex 3088

This theorem is used by:  r19.29af  3272  restmetu  24882  opreu2reuALT  33066  aciunf1lem  33249  fprodex01  33409  nsgqusf1olem1  33957  esplyfval1  34198  locfinreflem  34465  esumrnmpt2  34693  esum2dlem  34717  esum2d  34718  esumiun  34719