Theorem ssrexf 3980

Description: Restricted existential quantification follows from a subclass relationship. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
ssrexf.1	⊢ Ⅎ𝑥𝐴
ssrexf.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
ssrexf	⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))

Proof of Theorem ssrexf

Step	Hyp	Ref	Expression
1		ssrexf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2		ssrexf.2	. . . 4 ⊢ Ⅎ𝑥𝐵
3	1, 2	nfss 3906	. . 3 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵
4		ssel 3907	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
5	4	anim1d 623	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑)))
6	3, 5	eximd 2256	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
7		df-rex 3091	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
8		df-rex 3091	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
9	6, 7, 8	3imtr4g 299	1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1812   ∈ wcel 2147  Ⅎwnfc 2911  ∃wrex 3090   ⊆ wss 3900

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-10 2180  ax-11 2197  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-ss 3901

This theorem is used by:  iunxdif3  5058  stoweidlem34  47259