Theorem sbcreug 2838

Description: Interchange class substitution and restricted uniqueness quantifier. (Contributed by NM, 24-Feb-2013.)

Assertion

Ref	Expression
sbcreug	⊢ (𝐴 ∈ 𝑉 → ([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)   𝑉(𝑥,𝑦)

Proof of Theorem sbcreug

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2767	. 2 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑))
2		dfsbcq2 2767	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	reubidv 2493	. 2 ⊢ (𝑧 = 𝐴 → (∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
4		nfcv 2178	. . . 4 ⊢ Ⅎ𝑥𝐵
5		nfs1v 1815	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
6	4, 5	nfreuxy 2484	. . 3 ⊢ Ⅎ𝑥∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑
7		sbequ12 1654	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
8	7	reubidv 2493	. . 3 ⊢ (𝑥 = 𝑧 → (∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
9	6, 8	sbie 1674	. 2 ⊢ ([𝑧 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
10	1, 3, 9	vtoclbg 2614	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃!𝑦 ∈ 𝐵 𝜑 ↔ ∃!𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393  [wsb 1645  ∃!wreu 2308  [wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-reu 2313  df-v 2559  df-sbc 2765

This theorem is referenced by: (None)