Theorem ssrexf 3988

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 3915	. . 3 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵
4		ssel 3916	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
5	4	anim1d 617	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑)))
6	3, 5	eximd 2228	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
7		df-rex 3065	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
8		df-rex 3065	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
9	6, 7, 8	3imtr4g 297	1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1786   ∈ wcel 2119  Ⅎwnfc 2887  ∃wrex 3064   ⊆ wss 3890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-10 2152  ax-11 2168  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-ss 3907

This theorem is referenced by:  iunxdif3  5031  stoweidlem34  46484