Theorem ssrexv 3302

Description: Existential quantification restricted to a subclass. (Contributed by NM, 11-Jan-2007.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssrexv

Step	Hyp	Ref	Expression
1		ssel 3231	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	anim1d 336	. 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑)))
3	2	reximdv2 2641	1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ∈ wcel 2203  ∃wrex 2521   ⊆ wss 3210

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-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-rex 2526  df-in 3216  df-ss 3223

This theorem is referenced by:  iunss1  4001  moriotass  6033  tfr1onlemssrecs  6569  tfrcllemssrecs  6582  fiss  7263  supelti  7292  ctssdclemn0  7400  ctssdc  7403  enumctlemm  7404  nninfwlpoimlemginf  7466  ficardon  7484  rerecapb  9116  lbzbi  9947  zsupcl  10590  infssuzex  10592  fiubm  11191  rexico  11902  alzdvds  12536  bitsfzolem  12636  gcddvds  12655  dvdslegcd  12656  pclemub  12981  subrgdvds  14372  ssrest  15039  plyss  15595  reeff1olem  15628  bj-charfunbi  16573  bj-nn0suc  16726