Theorem reximd2a 2996

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of [Margaris] p. 90. (Contributed by Thierry Arnoux, 27-Jan-2020.)

Hypotheses

Ref	Expression
reximd2a.1	⊢ Ⅎ𝑥𝜑
reximd2a.2	⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝑥 ∈ 𝐵)
reximd2a.3	⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
reximd2a.4	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
reximd2a	⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜒)

Proof of Theorem reximd2a

Step	Hyp	Ref	Expression
1		reximd2a.4	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
2		reximd2a.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3		reximd2a.2	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝑥 ∈ 𝐵)
4		reximd2a.3	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
5	3, 4	jca 553	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → (𝑥 ∈ 𝐵 ∧ 𝜒))
6	5	expl 646	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) → (𝑥 ∈ 𝐵 ∧ 𝜒)))
7	2, 6	eximd 2072	. . 3 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜒)))
8		df-rex 2902	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))
9		df-rex 2902	. . 3 ⊢ (∃𝑥 ∈ 𝐵 𝜒 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜒))
10	7, 8, 9	3imtr4g 284	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐵 𝜒))
11	1, 10	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜒)

Syntax hints:   → wi 4   ∧ wa 383  ∃wex 1695  Ⅎwnf 1699   ∈ wcel 1977  ∃wrex 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701  df-rex 2902

This theorem is referenced by:  locfinreflem  29235