Theorem r19.29af2 3270

Description: A commonly used pattern based on r19.29 3125. (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 3269	. 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 1815   ∈ wcel 2144  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-nf 1816  df-ral 3077  df-rex 3087

This theorem is used by:  r19.29af  3271  restmetu  24898  opreu2reuALT  33124  aciunf1lem  33307  fprodex01  33466  nsgqusf1olem1  34014  esplyfval1  34255  locfinreflem  34522  esumrnmpt2  34750  esum2dlem  34774  esum2d  34775  esumiun  34776