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  6072  tfr1onlemssrecs  6614  tfrcllemssrecs  6627  fiss  7317  supelti  7349  ctssdclemn0  7457  ctssdc  7460  enumctlemm  7461  nninfwlpoimlemginf  7523  ficardon  7541  rerecapb  9182  lbzbi  10033  zsupcl  10682  infssuzex  10684  fiubm  11294  rexico  12011  alzdvds  12647  bitsfzolem  12747  gcddvds  12766  dvdslegcd  12767  nn0sqdcq  13014  pclemub  13096  subrgdvds  14653  ssrest  15420  plyss  15976  reeff1olem  16009  bj-charfunbi  17054  bj-nn0suc  17207