Theorem sbcrexgOLD 41826

Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcrex 3870 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦

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

Proof of Theorem sbcrexgOLD

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3781	. 2 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ [𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑))
2		dfsbcq2 3781	. . 3 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	rexbidv 3177	. 2 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
4		nfcv 2902	. . . 4 ⊢ Ⅎ𝑥𝐵
5		nfs1v 2152	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
6	4, 5	nfrexw 3309	. . 3 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑
7		sbequ12 2242	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
8	7	rexbidv 3177	. . 3 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
9	6, 8	sbie 2500	. 2 ⊢ ([𝑧 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
10	1, 3, 9	vtoclbg 3560	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1540  [wsb 2066   ∈ wcel 2105  ∃wrex 3069  [wsbc 3778

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-13 2370  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-sbc 3779

This theorem is referenced by:  2sbcrexOLD  41827  sbc2rexgOLD  41829