Theorem rexm 3564

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 2491	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
2		simpl 109	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴)
3	2	eximi 1624	. 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 1516   ∈ wcel 2177  ∃wrex 2486

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-rex 2491

This theorem is referenced by:  elrelimasn  5057  eusvobj2  5943  exmidomni  7259  fodjum  7263  ismgmid  13284  ismnd  13326  dfgrp2e  13435  zrhval  14454