Theorem rexlimd3 41781

Description: * Inference from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
rexlimd3.1	⊢ Ⅎ𝑥𝜑
rexlimd3.2	⊢ Ⅎ𝑥𝜒
rexlimd3.3	⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
rexlimd3	⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Proof of Theorem rexlimd3

Step	Hyp	Ref	Expression
1		rexlimd3.1	. 2 ⊢ Ⅎ𝑥𝜑
2		rexlimd3.2	. 2 ⊢ Ⅎ𝑥𝜒
3		rexlimd3.3	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
4	3	exp31 423	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
5	1, 2, 4	rexlimd 3276	1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 399  Ⅎwnf 1785   ∈ wcel 2111  ∃wrex 3107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786  df-ral 3111  df-rex 3112

This theorem is referenced by:  dffo3f  41806