Theorem ssrexf 3241

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 3172	. . 3 ⊢ Ⅎ𝑥 𝐴 ⊆ 𝐵
4		ssel 3173	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
5	4	anim1d 336	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑)))
6	3, 5	eximd 1623	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
7		df-rex 2478	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
8		df-rex 2478	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
9	6, 7, 8	3imtr4g 205	1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1503   ∈ wcel 2164  Ⅎwnfc 2323  ∃wrex 2473   ⊆ wss 3153

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-in 3159  df-ss 3166

This theorem is referenced by: (None)