Theorem rspesbca 3127

Description: Existence form of rspsbca 3126. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
rspesbca	⊢ ((A ∈ B ∧ [̣A / x]̣φ) → ∃x ∈ B φ)

Distinct variable group:   x,B

Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem rspesbca

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3050	. . 3 ⊢ (y = A → ([y / x]φ ↔ [̣A / x]̣φ))
2	1	rspcev 2956	. 2 ⊢ ((A ∈ B ∧ [̣A / x]̣φ) → ∃y ∈ B [y / x]φ)
3		cbvrexsv 2848	. 2 ⊢ (∃x ∈ B φ ↔ ∃y ∈ B [y / x]φ)
4	2, 3	sylibr 203	1 ⊢ ((A ∈ B ∧ [̣A / x]̣φ) → ∃x ∈ B φ)

Syntax hints:   → wi 4   ∧ wa 358  [wsb 1648   ∈ wcel 1710  ∃wrex 2616  [̣wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048

This theorem is referenced by:  spesbc  3128