Theorem sbcrext 3824

Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.) (Revised by NM, 18-Aug-2018.) (Proof shortened by JJ, 7-Jul-2021.)

Assertion

Ref	Expression
sbcrext	⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Distinct variable groups:   𝑥,𝑦   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑥, 𝑦)   𝐵(𝑦)

Proof of Theorem sbcrext

Step	Hyp	Ref	Expression
1		sbcex 3752	. . 3 ⊢ ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V)
2	1	a1i 11	. 2 ⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V))
3		nfnfc1 2929	. . 3 ⊢ Ⅎ𝑦Ⅎ𝑦𝐴
4		id 23	. . . 4 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦𝐴)
5		nfcvd 2927	. . . 4 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦V)
6	4, 5	nfeld 2937	. . 3 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦 𝐴 ∈ V)
7		sbcex 3752	. . . 4 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
8	7	2a1i 12	. . 3 ⊢ (Ⅎ𝑦𝐴 → (𝑦 ∈ 𝐵 → ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)))
9	3, 6, 8	rexlimd2 3272	. 2 ⊢ (Ⅎ𝑦𝐴 → (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V))
10		sbcng 3789	. . . . . 6 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑))
11	10	adantl 487	. . . . 5 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑))
12		sbcralt 3823	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑))
13	12	ancoms 464	. . . . . . 7 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑))
14	3, 6	nfan1 2240	. . . . . . . 8 ⊢ Ⅎ𝑦(Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V)
15		sbcng 3789	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
16	15	adantl 487	. . . . . . . 8 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
17	14, 16	ralbid 3279	. . . . . . 7 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → (∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
18	13, 17	bitrd 282	. . . . . 6 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
19	18	notbid 321	. . . . 5 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → (¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
20	11, 19	bitrd 282	. . . 4 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
21		dfrex2 3093	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑)
22	21	sbcbii 3798	. . . 4 ⊢ ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑)
23		dfrex2 3093	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑)
24	20, 22, 23	3bitr4g 317	. . 3 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
25	24	ex 418	. 2 ⊢ (Ⅎ𝑦𝐴 → (𝐴 ∈ V → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)))
26	2, 9, 25	pm5.21ndd 382	1 ⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  Ⅎwnfc 2911  ∀wral 3080  ∃wrex 3090  Vcvv 3454  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-v 3456  df-sbc 3743

This theorem is used by:  sbcrex  3826