Theorem sbcal 3825

Description: Move universal quantifier in and out of class substitution. (Contributed by NM, 31-Dec-2016.) (Revised by NM, 18-Aug-2018.)

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 3775	. 2 ⊢ ([𝐴 / 𝑦]∀𝑥𝜑 → 𝐴 ∈ V)
2		sbcex 3775	. . 3 ⊢ ([𝐴 / 𝑦]𝜑 → 𝐴 ∈ V)
3	2	sps 2185	. 2 ⊢ (∀𝑥[𝐴 / 𝑦]𝜑 → 𝐴 ∈ V)
4		dfsbcq2 3768	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ [𝐴 / 𝑦]∀𝑥𝜑))
5		dfsbcq2 3768	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑))
6	5	albidv 1920	. . 3 ⊢ (𝑧 = 𝐴 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
7		sbal 2169	. . 3 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
8	4, 6, 7	vtoclbg 3536	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
9	1, 3, 8	pm5.21nii 378	1 ⊢ ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)

Syntax hints:   ↔ wb 206  ∀wal 1538   = wceq 1540  [wsb 2064   ∈ wcel 2108  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-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-sbc 3766

This theorem is referenced by:  sbcabel  3853  sbcssg  4495  sbcfung  6560  bnj89  34752  bnj110  34889  bnj611  34949  bnj1000  34972  bj-sbeq  36919  bj-sbceqgALT  36920  sbcalf  38138  frege70  43957  frege77  43964  frege116  44003  frege118  44005  trsbc  44565  trsbcVD  44901  sbcssgVD  44907