Metamath Proof Explorer

Theorem reximdvai

Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.22 of Margaris p. 90. (Contributed by NM, 14-Nov-2002) Reduce dependencies on axioms. (Revised by Wolf Lammen, 8-Jan-2020)

Ref Expression
Hypothesis reximdvai.1 ${⊢}{\phi }\to \left({x}\in {A}\to \left({\psi }\to {\chi }\right)\right)$
Assertion reximdvai ${⊢}{\phi }\to \left(\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\psi }\to \exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\chi }\right)$

Proof

Step Hyp Ref Expression
1 reximdvai.1 ${⊢}{\phi }\to \left({x}\in {A}\to \left({\psi }\to {\chi }\right)\right)$
2 1 ralrimiv ${⊢}{\phi }\to \forall {x}\in {A}\phantom{\rule{.4em}{0ex}}\left({\psi }\to {\chi }\right)$
3 rexim ${⊢}\forall {x}\in {A}\phantom{\rule{.4em}{0ex}}\left({\psi }\to {\chi }\right)\to \left(\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\psi }\to \exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\chi }\right)$
4 2 3 syl ${⊢}{\phi }\to \left(\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\psi }\to \exists {x}\in {A}\phantom{\rule{.4em}{0ex}}{\chi }\right)$