Theorem 2sbcrex 43216

Description: Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by NM, 24-Aug-2018.)

Assertion

Ref	Expression
2sbcrex	⊢ ([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 2sbcrex

Step	Hyp	Ref	Expression
1		sbcrex 3813	. . 3 ⊢ ([𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐵 / 𝑏]𝜑)
2	1	sbcbii 3785	. 2 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ [𝐴 / 𝑎]∃𝑐 ∈ 𝐶 [𝐵 / 𝑏]𝜑)
3		sbcrex 3813	. 2 ⊢ ([𝐴 / 𝑎]∃𝑐 ∈ 𝐶 [𝐵 / 𝑏]𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)
4	2, 3	bitri 275	1 ⊢ ([𝐴 / 𝑎][𝐵 / 𝑏]∃𝑐 ∈ 𝐶 𝜑 ↔ ∃𝑐 ∈ 𝐶 [𝐴 / 𝑎][𝐵 / 𝑏]𝜑)

Syntax hints:   ↔ wb 206  ∃wrex 3061  [wsbc 3728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-v 3431  df-sbc 3729

This theorem is referenced by:  2rexfrabdioph  43224  4rexfrabdioph  43226