Theorem ssrexf 4062

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 3988	. . 3 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵
4		ssel 3989	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
5	4	anim1d 611	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑)))
6	3, 5	eximd 2214	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
7		df-rex 3069	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
8		df-rex 3069	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
9	6, 7, 8	3imtr4g 296	1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1776   ∈ wcel 2106  Ⅎwnfc 2888  ∃wrex 3068   ⊆ wss 3963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-10 2139  ax-11 2155  ax-12 2175

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-ss 3980

This theorem is referenced by:  iunxdif3  5100  stoweidlem34  45990