Theorem r19.29af2 2452

Description: A commonly used pattern based on r19.29 2450 (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 346	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
5	2, 4	ralrimi 2390	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒))
6	1, 5	jca 290	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)))
7		r19.29r 2451	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜒)) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ (𝜓 → 𝜒)))
8		r19.29af2.c	. . 3 ⊢ Ⅎ𝑥𝜒
9		pm3.35 329	. . . 4 ⊢ ((𝜓 ∧ (𝜓 → 𝜒)) → 𝜒)
10	9	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((𝜓 ∧ (𝜓 → 𝜒)) → 𝜒))
11	8, 10	rexlimi 2426	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜓 ∧ (𝜓 → 𝜒)) → 𝜒)
12	6, 7, 11	3syl 17	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 97  Ⅎwnf 1349   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-ral 2311  df-rex 2312

This theorem is referenced by:  r19.29af  2453