Theorem ssrexv 3292

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 3221	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	anim1d 336	. 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑)))
3	2	reximdv2 2631	1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ∈ wcel 2202  ∃wrex 2511   ⊆ wss 3200

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-rex 2516  df-in 3206  df-ss 3213

This theorem is referenced by:  iunss1  3981  moriotass  6002  tfr1onlemssrecs  6505  tfrcllemssrecs  6518  fiss  7176  supelti  7201  ctssdclemn0  7309  ctssdc  7312  enumctlemm  7313  nninfwlpoimlemginf  7375  ficardon  7393  rerecapb  9023  lbzbi  9850  zsupcl  10491  infssuzex  10493  fiubm  11092  rexico  11782  alzdvds  12416  bitsfzolem  12516  gcddvds  12535  dvdslegcd  12536  pclemub  12861  subrgdvds  14251  ssrest  14908  plyss  15464  reeff1olem  15497  bj-charfunbi  16409  bj-nn0suc  16562