Theorem sbcreu 3764

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 3693	. 2 ⊢ ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 → 𝐴 ∈ V)
2		reurex 3371	. . 3 ⊢ (∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)
3		sbcex 3693	. . . 4 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
4	3	rexlimivw 3227	. . 3 ⊢ (∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
5	2, 4	syl 17	. 2 ⊢ (∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
6		dfsbcq2 3686	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑))
7		dfsbcq2 3686	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
8	7	reubidv 3329	. . 3 ⊢ (𝑧 = 𝐴 → (∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
9		nfcv 2932	. . . . 5 ⊢ Ⅎ𝑥𝐵
10		nfs1v 2202	. . . . 5 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
11	9, 10	nfreu 3315	. . . 4 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑
12		sbequ12 2179	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
13	12	reubidv 3329	. . . 4 ⊢ (𝑥 = 𝑧 → (∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
14	11, 13	sbie 2468	. . 3 ⊢ ([𝑧 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
15	6, 8, 14	vtoclbg 3487	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
16	1, 5, 15	pm5.21nii 371	1 ⊢ ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 198   = wceq 1507  [wsb 2015   ∈ wcel 2050  ∃wrex 3089  ∃!wreu 3090  Vcvv 3415  [wsbc 3683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-v 3417  df-sbc 3684

This theorem is referenced by: (None)