Theorem spesbc 3128

Description: Existence form of spsbc 3059. (Contributed by Mario Carneiro, 18-Nov-2016.)

Assertion

Ref	Expression
spesbc	⊢ ([̣A / x]̣φ → ∃xφ)

Proof of Theorem spesbc

Step	Hyp	Ref	Expression
1		sbcex 3056	. . 3 ⊢ ([̣A / x]̣φ → A ∈ V)
2		rspesbca 3127	. . 3 ⊢ ((A ∈ V ∧ [̣A / x]̣φ) → ∃x ∈ V φ)
3	1, 2	mpancom 650	. 2 ⊢ ([̣A / x]̣φ → ∃x ∈ V φ)
4		rexv 2874	. 2 ⊢ (∃x ∈ V φ ↔ ∃xφ)
5	3, 4	sylib 188	1 ⊢ ([̣A / x]̣φ → ∃xφ)

This proof depends on syntax axioms:   → wi 4  ∃wex 1541   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860  [̣wsbc 3047

This proof depends on 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 proof 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 used by:  spesbcd  3129  opelopabsb  4698