Theorem sbcex2 3802

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 3751	. 2 ⊢ ([𝐴 / 𝑦]∃𝑥𝜑 → 𝐴 ∈ V)
2		sbcex 3751	. . 3 ⊢ ([𝐴 / 𝑦]𝜑 → 𝐴 ∈ V)
3	2	exlimiv 1931	. 2 ⊢ (∃𝑥[𝐴 / 𝑦]𝜑 → 𝐴 ∈ V)
4		dfsbcq2 3744	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∃𝑥𝜑 ↔ [𝐴 / 𝑦]∃𝑥𝜑))
5		dfsbcq2 3744	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑))
6	5	exbidv 1922	. . 3 ⊢ (𝑧 = 𝐴 → (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
7		sbex 2283	. . 3 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
8	4, 6, 7	vtoclbg 3512	. 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   = wceq 1541  ∃wex 1780  [wsb 2067   ∈ wcel 2111  Vcvv 3436  [wsbc 3741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-sbc 3742

This theorem is referenced by:  sbcabel  3829  csbuni  4888  csbxp  5716  csbdm  5837  sbcfung  6505  csbfrecsg  8214  bnj89  34731  bnj985v  34963  bnj985  34964  csboprabg  37370  sbcexf  38161  onfrALTlem5  44581  onfrALTlem5VD  44923  csbxpgVD  44932  csbrngVD  44934  csbunigVD  44936