Theorem sbcexg 3005

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

Assertion

Ref	Expression
sbcexg	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦

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

Proof of Theorem sbcexg

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2954	. 2 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∃𝑥𝜑 ↔ [𝐴 / 𝑦]∃𝑥𝜑))
2		dfsbcq2 2954	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑))
3	2	exbidv 1813	. 2 ⊢ (𝑧 = 𝐴 → (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
4		sbex 1992	. 2 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
5	1, 3, 4	vtoclbg 2787	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343  ∃wex 1480  [wsb 1750   ∈ wcel 2136  [wsbc 2951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952

This theorem is referenced by:  sbcabel  3032  csbunig  3797  csbxpg  4685  csbrng  5065