Theorem sbc2rexgOLD 42730

Description: Exchange a substitution with two existentials. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7469 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
sbc2rexgOLD	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎]𝜑))

Distinct variable groups:   𝐴,𝑏   𝐴,𝑐   𝐵,𝑎   𝐶,𝑎   𝑎,𝑏   𝑎,𝑐

Allowed substitution hints:   𝜑(𝑎,𝑏,𝑐)   𝐴(𝑎)   𝐵(𝑏,𝑐)   𝐶(𝑏,𝑐)   𝑉(𝑎,𝑏,𝑐)

Proof of Theorem sbc2rexgOLD

Step	Hyp	Ref	Expression
1		elex 3498	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		sbcrexgOLD 42727	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑏 ∈ 𝐵 [𝐴 / 𝑎]∃𝑐 ∈ 𝐶 𝜑))
3		sbcrexgOLD 42727	. . . 4 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑎]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎]𝜑))
4	3	rexbidv 3175	. . 3 ⊢ (𝐴 ∈ V → (∃𝑏 ∈ 𝐵 [𝐴 / 𝑎]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎]𝜑))
5	2, 4	bitrd 279	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎]𝜑))
6	1, 5	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑎]∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑏 ∈ 𝐵 ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎]𝜑))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2104  ∃wrex 3066  Vcvv 3477  [wsbc 3791

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 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-13 2373  ax-ext 2704

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1538  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ral 3058  df-rex 3067  df-v 3479  df-sbc 3792

This theorem is referenced by:  sbc4rexgOLD  42732