Theorem rexm 3550

Description: Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)

Assertion

Ref	Expression
rexm	⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexm

Step	Hyp	Ref	Expression
1		df-rex 2481	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		simpl 109	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴)
3	2	eximi 1614	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥 𝑥 ∈ 𝐴)
4	1, 3	sylbi 121	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1506   ∈ wcel 2167  ∃wrex 2476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548

This theorem depends on definitions:  df-bi 117  df-rex 2481

This theorem is referenced by:  elrelimasn  5035  eusvobj2  5908  exmidomni  7208  fodjum  7212  ismgmid  13020  ismnd  13060  dfgrp2e  13160  zrhval  14173