Theorem sbcex2 3781

Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
sbcex2	⊢ ([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 sbcex2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcex 3726	. 2 ⊢ ([𝐴 / 𝑦]∃𝑥𝜑 → 𝐴 ∈ V)
2		sbcex 3726	. . 3 ⊢ ([𝐴 / 𝑦]𝜑 → 𝐴 ∈ V)
3	2	exlimiv 1933	. 2 ⊢ (∃𝑥[𝐴 / 𝑦]𝜑 → 𝐴 ∈ V)
4		dfsbcq2 3719	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∃𝑥𝜑 ↔ [𝐴 / 𝑦]∃𝑥𝜑))
5		dfsbcq2 3719	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑))
6	5	exbidv 1924	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
7		sbex 2278	. . 3 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
8	4, 6, 7	vtoclbg 3507	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
9	1, 3, 8	pm5.21nii 380	1 ⊢ ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑)

Syntax hints:   ↔ wb 205   = wceq 1539  ∃wex 1782  [wsb 2067   ∈ wcel 2106  Vcvv 3432  [wsbc 3716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-sbc 3717

This theorem is referenced by:  sbcabel  3811  csbuni  4870  csbxp  5686  csbdm  5806  sbcfung  6458  csbfrecsg  8100  bnj89  32700  bnj985v  32933  bnj985  32934  csboprabg  35501  sbcexf  36273  onfrALTlem5  42162  onfrALTlem5VD  42505  csbxpgVD  42514  csbrngVD  42516  csbunigVD  42518