Theorem r19.29af 3274

Description: A commonly used pattern based on r19.29 3120. See r19.29a 3168, r19.29an 3164 for a variant when 𝑥 is disjoint from 𝜑. (Contributed by Thierry Arnoux, 29-Nov-2017.)

Hypotheses

Ref	Expression
r19.29af.0	⊢ Ⅎ𝑥𝜑
r19.29af.1	⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
r19.29af.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
r19.29af	⊢ (𝜑 → 𝜒)

Distinct variable group:   𝜒,𝑥

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

Proof of Theorem r19.29af

Step	Hyp	Ref	Expression
1		r19.29af.0	. 2 ⊢ Ⅎ𝑥𝜑
2		nfv 1913	. 2 ⊢ Ⅎ𝑥𝜒
3		r19.29af.1	. 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
4		r19.29af.2	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
5	1, 2, 3, 4	r19.29af2 3273	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:  fsnex  7319  neiptopnei  23161  neitr  23209  utopsnneiplem  24277  isucn2  24309  2sqmo  27499  foresf1o  32532  fsumiunle  32833  nsgqusf1olem3  33408  irngnzply1  33691  reff  33785  locfinreflem  33786  ordtconnlem1  33870  esumrnmpt2  34032  esumgect  34054  esum2dlem  34056  esum2d  34057  esumiun  34058  sigapildsys  34126  oms0  34262  eulerpartlemgvv  34341  breprexplema  34607  stoweidlem27  45948  stoweidlem35  45956