Theorem sbcal 3485

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 3445	. 2 ⊢ ([𝐴 / 𝑦]∀𝑥𝜑 → 𝐴 ∈ V)
2		sbcex 3445	. . 3 ⊢ ([𝐴 / 𝑦]𝜑 → 𝐴 ∈ V)
3	2	sps 2055	. 2 ⊢ (∀𝑥[𝐴 / 𝑦]𝜑 → 𝐴 ∈ V)
4		dfsbcq2 3438	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∀𝑥𝜑 ↔ [𝐴 / 𝑦]∀𝑥𝜑))
5		dfsbcq2 3438	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑))
6	5	albidv 1849	. . 3 ⊢ (𝑧 = 𝐴 → (∀𝑥[𝑧 / 𝑦]𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
7		sbal 2462	. . 3 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
8	4, 6, 7	vtoclbg 3267	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑))
9	1, 3, 8	pm5.21nii 368	1 ⊢ ([𝐴 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝐴 / 𝑦]𝜑)

Syntax hints:   ↔ wb 196  ∀wal 1481   = wceq 1483  [wsb 1880   ∈ wcel 1990  Vcvv 3200  [wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  sbcabel  3517  sbcssg  4085  sbcfung  5912  bnj89  30787  bnj538OLD  30810  bnj110  30928  bnj611  30988  bnj1000  31011  bj-sbeq  32896  bj-sbceqgALT  32897  sbcalf  33917  frege70  38227  frege77  38234  frege116  38273  frege118  38275  trsbc  38750