Theorem spcimedv 2816

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

Hypotheses

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

Assertion

Ref	Expression
spcimedv	⊢ (𝜑 → (𝜒 → ∃𝑥𝜓))

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

Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem spcimedv

Step	Hyp	Ref	Expression
1		spcimedv.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))
2	1	ex 114	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜒 → 𝜓)))
3	2	alrimiv 1867	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜒 → 𝜓)))
4		spcimdv.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
5		nfv 1521	. . 3 ⊢ Ⅎ𝑥𝜒
6		nfcv 2312	. . 3 ⊢ Ⅎ𝑥𝐴
7	5, 6	spcimegft 2808	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜒 → 𝜓)) → (𝐴 ∈ 𝐵 → (𝜒 → ∃𝑥𝜓)))
8	3, 4, 7	sylc 62	1 ⊢ (𝜑 → (𝜒 → ∃𝑥𝜓))

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1346   = wceq 1348  ∃wex 1485   ∈ wcel 2141

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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  rspcimedv  2836  fihashf1rn  10723