Theorem sbcrext 3802

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 3721	. . 3 ⊢ ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V)
2	1	a1i 11	. 2 ⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V))
3		nfnfc1 2909	. . 3 ⊢ Ⅎ𝑦Ⅎ𝑦𝐴
4		id 22	. . . 4 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦𝐴)
5		nfcvd 2907	. . . 4 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦V)
6	4, 5	nfeld 2917	. . 3 ⊢ (Ⅎ𝑦𝐴 → Ⅎ𝑦 𝐴 ∈ V)
7		sbcex 3721	. . . 4 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
8	7	2a1i 12	. . 3 ⊢ (Ⅎ𝑦𝐴 → (𝑦 ∈ 𝐵 → ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)))
9	3, 6, 8	rexlimd2 3244	. 2 ⊢ (Ⅎ𝑦𝐴 → (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V))
10		sbcng 3761	. . . . . 6 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑))
11	10	adantl 481	. . . . 5 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑))
12		sbcralt 3801	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ Ⅎ𝑦𝐴) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑))
13	12	ancoms 458	. . . . . . 7 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑))
14	3, 6	nfan1 2196	. . . . . . . 8 ⊢ Ⅎ𝑦(Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V)
15		sbcng 3761	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
16	15	adantl 481	. . . . . . . 8 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ 𝜑 ↔ ¬ [𝐴 / 𝑥]𝜑))
17	14, 16	ralbid 3158	. . . . . . 7 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → (∀𝑦 ∈ 𝐵 [𝐴 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
18	13, 17	bitrd 278	. . . . . 6 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
19	18	notbid 317	. . . . 5 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → (¬ [𝐴 / 𝑥]∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
20	11, 19	bitrd 278	. . . 4 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑))
21		dfrex2 3166	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑)
22	21	sbcbii 3772	. . . 4 ⊢ ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥] ¬ ∀𝑦 ∈ 𝐵 ¬ 𝜑)
23		dfrex2 3166	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ [𝐴 / 𝑥]𝜑)
24	20, 22, 23	3bitr4g 313	. . 3 ⊢ ((Ⅎ𝑦𝐴 ∧ 𝐴 ∈ V) → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
25	24	ex 412	. 2 ⊢ (Ⅎ𝑦𝐴 → (𝐴 ∈ V → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)))
26	2, 9, 25	pm5.21ndd 380	1 ⊢ (Ⅎ𝑦𝐴 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  Ⅎwnfc 2886  ∀wral 3063  ∃wrex 3064  Vcvv 3422  [wsbc 3711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-v 3424  df-sbc 3712

This theorem is referenced by:  sbcrex  3804