Theorem spesbc 3828

Description: Existence form of spsbc 3751. (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 3748	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
2		rspesbca 3827	. . 3 ⊢ ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑)
3	1, 2	mpancom 701	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑)
4		rexv 3477	. 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
5	3, 4	sylib 221	1 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)

This proof depends on syntax axioms:   → wi 4  ∃wex 1811   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450  [wsbc 3738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-v 3452  df-sbc 3739

This theorem is used by:  spesbcd  3829  opelopabsb  5503  sbccomieg  43894  frege124d  44855  sbiota1  45512