Theorem 2sbcrexOLD 39373

Description: Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 7190 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 39372	. . . 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 3827	. 2 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ [𝐴 / 𝑎]∃𝑐 ∈ 𝐶 [𝐵 / 𝑏]𝜑)
5		2sbcrex.1	. . 3 ⊢ 𝐴 ∈ V
6		sbcrexgOLD 39372	. . 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 277	1 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)

Syntax hints:   ↔ wb 208   ∈ wcel 2107  ∃wrex 3137  Vcvv 3493  [wsbc 3770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2383  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-sbc 3771

This theorem is referenced by: (None)