Theorem ssrexv 3260

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 3189	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	anim1d 336	. 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑)))
3	2	reximdv2 2606	1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ∈ wcel 2177  ∃wrex 2486   ⊆ wss 3168

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-rex 2491  df-in 3174  df-ss 3181

This theorem is referenced by:  iunss1  3941  moriotass  5938  tfr1onlemssrecs  6435  tfrcllemssrecs  6448  fiss  7091  supelti  7116  ctssdclemn0  7224  ctssdc  7227  enumctlemm  7228  nninfwlpoimlemginf  7290  ficardon  7308  rerecapb  8929  lbzbi  9750  zsupcl  10387  infssuzex  10389  fiubm  10986  rexico  11582  alzdvds  12215  bitsfzolem  12315  gcddvds  12334  dvdslegcd  12335  pclemub  12660  subrgdvds  14047  ssrest  14704  plyss  15260  reeff1olem  15293  bj-charfunbi  15861  bj-nn0suc  16014