Theorem sbcreu 3851

Description: Interchange class substitution and restricted unique existential quantifier. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
sbcreu	⊢ ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)

Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦

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

Proof of Theorem sbcreu

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3775	. 2 ⊢ ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V)
2		reurex 3363	. . 3 ⊢ (∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)
3		sbcex 3775	. . . 4 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
4	3	rexlimivw 3137	. . 3 ⊢ (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
5	2, 4	syl 17	. 2 ⊢ (∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
6		dfsbcq2 3768	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑))
7		dfsbcq2 3768	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
8	7	reubidv 3377	. . 3 ⊢ (𝑧 = 𝐴 → (∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
9		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥𝐵
10		nfs1v 2156	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
11	9, 10	nfreuw 3393	. . . 4 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑
12		sbequ12 2251	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
13	12	reubidv 3377	. . . 4 ⊢ (𝑥 = 𝑧 → (∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
14	11, 13	sbiev 2314	. . 3 ⊢ ([𝑧 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
15	6, 8, 14	vtoclbg 3536	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
16	1, 5, 15	pm5.21nii 378	1 ⊢ ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 206   = wceq 1540  [wsb 2064   ∈ wcel 2108  ∃wrex 3060  ∃!wreu 3357  Vcvv 3459  [wsbc 3765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-rex 3061  df-rmo 3359  df-reu 3360  df-v 3461  df-sbc 3766

This theorem is referenced by: (None)