Metamath Proof Explorer

Theorem bnj984

Description: Technical lemma for bnj69 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj984.3 ${⊢}{\chi }↔\left({n}\in {D}\wedge {f}Fn{n}\wedge {\phi }\wedge {\psi }\right)$
bnj984.11 ${⊢}{B}=\left\{{f}|\exists {n}\in {D}\phantom{\rule{.4em}{0ex}}\left({f}Fn{n}\wedge {\phi }\wedge {\psi }\right)\right\}$
Assertion bnj984

Proof

Step Hyp Ref Expression
1 bnj984.3 ${⊢}{\chi }↔\left({n}\in {D}\wedge {f}Fn{n}\wedge {\phi }\wedge {\psi }\right)$
2 bnj984.11 ${⊢}{B}=\left\{{f}|\exists {n}\in {D}\phantom{\rule{.4em}{0ex}}\left({f}Fn{n}\wedge {\phi }\wedge {\psi }\right)\right\}$
3 sbc8g
4 2 eleq2i ${⊢}{G}\in {B}↔{G}\in \left\{{f}|\exists {n}\in {D}\phantom{\rule{.4em}{0ex}}\left({f}Fn{n}\wedge {\phi }\wedge {\psi }\right)\right\}$
5 3 4 syl6rbbr
6 df-rex ${⊢}\exists {n}\in {D}\phantom{\rule{.4em}{0ex}}\left({f}Fn{n}\wedge {\phi }\wedge {\psi }\right)↔\exists {n}\phantom{\rule{.4em}{0ex}}\left({n}\in {D}\wedge \left({f}Fn{n}\wedge {\phi }\wedge {\psi }\right)\right)$
7 bnj252 ${⊢}\left({n}\in {D}\wedge {f}Fn{n}\wedge {\phi }\wedge {\psi }\right)↔\left({n}\in {D}\wedge \left({f}Fn{n}\wedge {\phi }\wedge {\psi }\right)\right)$
8 1 7 bitri ${⊢}{\chi }↔\left({n}\in {D}\wedge \left({f}Fn{n}\wedge {\phi }\wedge {\psi }\right)\right)$
9 6 8 bnj133 ${⊢}\exists {n}\in {D}\phantom{\rule{.4em}{0ex}}\left({f}Fn{n}\wedge {\phi }\wedge {\psi }\right)↔\exists {n}\phantom{\rule{.4em}{0ex}}{\chi }$
10 9 sbcbii
11 5 10 syl6bb