Theorem sbc4rexgOLD 42182

Description: Exchange a substitution with four existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7456 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbc4rexgOLD	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 [𝐴 / 𝑎]𝜑))

Distinct variable groups:   𝐴,𝑏   𝐴,𝑐   𝐵,𝑎   𝐶,𝑎   𝑎,𝑏   𝑎,𝑐   𝐴,𝑑   𝐴,𝑒   𝐷,𝑎   𝐸,𝑎   𝑎,𝑑   𝑒,𝑎

Allowed substitution hints:   𝜑(𝑒,𝑎,𝑏,𝑐,𝑑)   𝐴(𝑎)   𝐵(𝑒,𝑏,𝑐,𝑑)   𝐶(𝑒,𝑏,𝑐,𝑑)   𝐷(𝑒,𝑏,𝑐,𝑑)   𝐸(𝑒,𝑏,𝑐,𝑑)   𝑉(𝑒,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem sbc4rexgOLD

Step	Hyp	Ref	Expression
1		elex 3488	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		sbc2rexgOLD 42180	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎]∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 𝜑))
3		sbc2rexgOLD 42180	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑎]∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 𝜑 ↔ ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 [𝐴 / 𝑎]𝜑))
4	3	2rexbidv 3214	. . 3 ⊢ (𝐴 ∈ V → (∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎]∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 [𝐴 / 𝑎]𝜑))
5	2, 4	bitrd 279	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 [𝐴 / 𝑎]𝜑))
6	1, 5	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 ∃𝑑 ∈ 𝐷 ∃𝑒 ∈ 𝐸 [𝐴 / 𝑎]𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2099  ∃wrex 3065  Vcvv 3469  [wsbc 3774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-13 2366  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-v 3471  df-sbc 3775

This theorem is referenced by: (None)