Theorem r19.29af2 2606

Description: A commonly used pattern based on r19.29 2603. (Contributed by Thierry Arnoux, 17-Dec-2017.)

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	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
2		r19.29af2.p	. . . 4 ⊢ Ⅎ𝑥𝜑
3		r19.29af2.1	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
4	3	exp31 362	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
5	2, 4	ralrimi 2537	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
6	1, 5	jca 304	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)))
7		r19.29r 2604	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ (𝜓 → 𝜒)))
8		r19.29af2.c	. . 3 ⊢ Ⅎ𝑥𝜒
9		pm3.35 345	. . . 4 ⊢ ((𝜓 ∧ (𝜓 → 𝜒)) → 𝜒)
10	9	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((𝜓 ∧ (𝜓 → 𝜒)) → 𝜒))
11	8, 10	rexlimi 2576	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ (𝜓 → 𝜒)) → 𝜒)
12	6, 7, 11	3syl 17	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 103  Ⅎwnf 1448   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-ral 2449  df-rex 2450

This theorem is referenced by:  r19.29af  2607  ctiunctlemfo  12372