Theorem ssrexv 3293

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 3222	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	anim1d 336	. 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑)))
3	2	reximdv2 2632	1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ∈ wcel 2202  ∃wrex 2512   ⊆ wss 3201

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 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-rex 2517  df-in 3207  df-ss 3214

This theorem is referenced by:  iunss1  3986  moriotass  6012  tfr1onlemssrecs  6548  tfrcllemssrecs  6561  fiss  7219  supelti  7244  ctssdclemn0  7352  ctssdc  7355  enumctlemm  7356  nninfwlpoimlemginf  7418  ficardon  7436  rerecapb  9066  lbzbi  9893  zsupcl  10535  infssuzex  10537  fiubm  11136  rexico  11842  alzdvds  12476  bitsfzolem  12576  gcddvds  12595  dvdslegcd  12596  pclemub  12921  subrgdvds  14311  ssrest  14973  plyss  15529  reeff1olem  15562  bj-charfunbi  16507  bj-nn0suc  16660