Theorem r19.29af2 3273

Description: A commonly used pattern based on r19.29 3120. (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 419	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
6	2, 3, 5	rexlimd 3272	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))
7	1, 6	mpd 15	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 395  Ⅎwnf 1781   ∈ wcel 2108  ∃wrex 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782  df-ral 3068  df-rex 3077

This theorem is referenced by:  r19.29af  3274  restmetu  24604  opreu2reuALT  32505  aciunf1lem  32680  fprodex01  32829  nsgqusf1olem1  33406  locfinreflem  33786  esumrnmpt2  34032  esum2dlem  34056  esum2d  34057  esumiun  34058