Theorem exellimddv 32369

Description: Eliminate an antecedent when the antecedent is elementhood, deduction version. See exellim 32368 for the closed form, which requires the use of a universal quantifier. (Contributed by ML, 17-Jul-2020.)

Hypotheses

Ref	Expression
exellimddv.1	⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
exellimddv.2	⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))

Assertion

Ref	Expression
exellimddv	⊢ (𝜑 → 𝜓)

Distinct variable groups:   𝜑,𝑥   𝜓,𝑥

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem exellimddv

Step	Hyp	Ref	Expression
1		exellimddv.1	. 2 ⊢ (𝜑 → ∃𝑥 𝑥 ∈ 𝐴)
2		exellimddv.2	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
3	2	alrimiv 1842	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))
4		exellim 32368	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) → 𝜓)
5	1, 3, 4	syl2anc 691	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695   ∈ wcel 1977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701

This theorem is referenced by:  topdifinffinlem  32371