Theorem 2sbcrexOLD 40605

Description: Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7313 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
2sbcrex.1	⊢ 𝐴 ∈ V
2sbcrex.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
2sbcrexOLD	⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)

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

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

Proof of Theorem 2sbcrexOLD

Step	Hyp	Ref	Expression
1		2sbcrex.2	. . . 4 ⊢ 𝐵 ∈ V
2		sbcrexgOLD 40604	. . . 4 ⊢ (𝐵 ∈ V → ([𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐵 / 𝑏]𝜑))
3	1, 2	ax-mp 5	. . 3 ⊢ ([𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐵 / 𝑏]𝜑)
4	3	sbcbii 3781	. 2 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ [𝐴 / 𝑎]∃𝑐 ∈ 𝐶 [𝐵 / 𝑏]𝜑)
5		2sbcrex.1	. . 3 ⊢ 𝐴 ∈ V
6		sbcrexgOLD 40604	. . 3 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑎]∃𝑐 ∈ 𝐶 [𝐵 / 𝑏]𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑))
7	5, 6	ax-mp 5	. 2 ⊢ ([𝐴 / 𝑎]∃𝑐 ∈ 𝐶 [𝐵 / 𝑏]𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)
8	4, 7	bitri 274	1 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)

Syntax hints:   ↔ wb 205   ∈ wcel 2110  ∃wrex 3067  Vcvv 3431  [wsbc 3720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-13 2374  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-sbc 3721

This theorem is referenced by: (None)