Theorem sbcal 3080

Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.)

Assertion

Ref	Expression
sbcal	⊢ ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦

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

Proof of Theorem sbcal

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3037	. 2 ⊢ ([𝐴 / 𝑦]∀𝑥𝜑 → 𝐴 ∈ V)
2		sbcex 3037	. . 3 ⊢ ([𝐴 / 𝑦]𝜑 → 𝐴 ∈ V)
3	2	sps 1583	. 2 ⊢ (∀𝑥[𝐴 / 𝑦]𝜑 → 𝐴 ∈ V)
4		dfsbcq2 3031	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ [𝐴 / 𝑦]∀𝑥𝜑))
5		dfsbcq2 3031	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑))
6	5	albidv 1870	. . 3 ⊢ (𝑧 = 𝐴 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
7		sbal 2051	. . 3 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
8	4, 6, 7	vtoclbg 2862	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
9	1, 3, 8	pm5.21nii 709	1 ⊢ ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)

Syntax hints:   ↔ wb 105  ∀wal 1393   = wceq 1395  [wsb 1808   ∈ wcel 2200  Vcvv 2799  [wsbc 3028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029

This theorem is referenced by:  sbcfung  5341