Theorem ssrexv 3307

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 3236	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	anim1d 336	. 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑)))
3	2	reximdv2 2643	1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ∈ wcel 2205  ∃wrex 2523   ⊆ wss 3214

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 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-rex 2528  df-in 3220  df-ss 3227

This theorem is referenced by:  iunss1  4007  moriotass  6042  tfr1onlemssrecs  6583  tfrcllemssrecs  6596  fiss  7277  supelti  7306  ctssdclemn0  7414  ctssdc  7417  enumctlemm  7418  nninfwlpoimlemginf  7480  ficardon  7498  rerecapb  9137  lbzbi  9969  zsupcl  10616  infssuzex  10618  fiubm  11223  rexico  11935  alzdvds  12569  bitsfzolem  12669  gcddvds  12688  dvdslegcd  12689  pclemub  13014  subrgdvds  14485  ssrest  15177  plyss  15733  reeff1olem  15766  bj-charfunbi  16721  bj-nn0suc  16874