Theorem ssrexv 3249

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 3178	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
2	1	anim1d 336	. 2 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜑)))
3	2	reximdv2 2596	1 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 → ∃𝑥 ∈ 𝐵 𝜑))

Syntax hints:   → wi 4   ∈ wcel 2167  ∃wrex 2476   ⊆ wss 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-rex 2481  df-in 3163  df-ss 3170

This theorem is referenced by:  iunss1  3928  moriotass  5909  tfr1onlemssrecs  6406  tfrcllemssrecs  6419  fiss  7052  supelti  7077  ctssdclemn0  7185  ctssdc  7188  enumctlemm  7189  nninfwlpoimlemginf  7251  ficardon  7267  rerecapb  8887  lbzbi  9707  zsupcl  10338  infssuzex  10340  fiubm  10937  rexico  11403  alzdvds  12036  bitsfzolem  12136  gcddvds  12155  dvdslegcd  12156  pclemub  12481  subrgdvds  13867  ssrest  14502  plyss  15058  reeff1olem  15091  bj-charfunbi  15541  bj-nn0suc  15694