Theorem sbcexgOLD 38753

Description: Move existential quantifier in and out of class substitution. (Contributed by NM, 21-May-2004.) Obsolete as of 17-Aug-2018. Use sbcex 3445 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbcexgOLD	⊢ (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 sbcexgOLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3438	. 2 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]∃𝑥𝜑 ↔ [𝐴 / 𝑦]∃𝑥𝜑))
2		dfsbcq2 3438	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑦]𝜑))
3	2	exbidv 1850	. 2 ⊢ (𝑧 = 𝐴 → (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))
4		sbex 2463	. 2 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
5	1, 3, 4	vtoclbg 3267	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝐴 / 𝑦]𝜑))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483  ∃wex 1704  [wsb 1880   ∈ wcel 1990  [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:  csbunigOLD  39051  csbxpgOLD  39053  csbrngOLD  39056  onfrALTlem5VD  39121  csbxpgVD  39130  csbrngVD  39132  csbunigVD  39134