Theorem rspcimedv 3284

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	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		rspcimedv.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))
3	2	con3d 147	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒))
4	1, 3	rspcimdv 3283	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ¬ 𝜓 → ¬ 𝜒))
5	4	con2d 128	. 2 ⊢ (𝜑 → (𝜒 → ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜓))
6		dfrex2 2979	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜓)
7	5, 6	syl6ibr 241	1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175

This theorem is referenced by:  rspcedv  3286  scshwfzeqfzo  13423  symgfixfo  17682  slesolex  20307  clwlkfoclwwlk  26372  el2wlkonot  26396  el2spthonot  26397  el2wlkonotot0  26399  usg2spot2nb  26592  usgr2pthlem  40969  clwlksfoclwwlk  41270  fusgr2wsp2nb  41498