Theorem rspcimedv 2673

Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
rspcimdv.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
rspcimedv.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcimedv

Step	Hyp	Ref	Expression
1		rspcimdv.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		simpr 107	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
3	2	eleq1d 2120	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
4	3	biimprd 151	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐴 ∈ 𝐵 → 𝑥 ∈ 𝐵))
5		rspcimedv.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))
6	4, 5	anim12d 322	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐴 ∈ 𝐵 ∧ 𝜒) → (𝑥 ∈ 𝐵 ∧ 𝜓)))
7	1, 6	spcimedv 2654	. . 3 ⊢ (𝜑 → ((𝐴 ∈ 𝐵 ∧ 𝜒) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)))
8	1, 7	mpand 413	. 2 ⊢ (𝜑 → (𝜒 → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)))
9		df-rex 2327	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜓))
10	8, 9	syl6ibr 155	1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))

Syntax hints:   → wi 4   ∧ wa 101   = wceq 1257  ∃wex 1395   ∈ wcel 1407  ∃wrex 2322

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036

This theorem depends on definitions:  df-bi 114  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-rex 2327  df-v 2574

This theorem is referenced by:  rspcedv  2675