Theorem ssrexv 3313

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 3242	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	anim1d 336	. 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑)))
3	2	reximdv2 2649	1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2209  ∃wrex 2529   ⊆ wss 3220

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-rex 2534  df-in 3226  df-ss 3233

This theorem is used by:  iunss1  4023  moriotass  6069  tfr1onlemssrecs  6610  tfrcllemssrecs  6623  fiss  7311  supelti  7343  ctssdclemn0  7451  ctssdc  7454  enumctlemm  7455  nninfwlpoimlemginf  7517  ficardon  7535  rerecapb  9176  lbzbi  10026  zsupcl  10675  infssuzex  10677  fiubm  11286  rexico  12003  alzdvds  12639  bitsfzolem  12739  gcddvds  12758  dvdslegcd  12759  nn0sqdcq  13006  pclemub  13088  subrgdvds  14594  ssrest  15335  plyss  15891  reeff1olem  15924  bj-charfunbi  16959  bj-nn0suc  17112