Theorem spesbc 3857

Description: Existence form of spsbc 3778. (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 3775	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
2		rspesbca 3856	. . 3 ⊢ ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑)
3	1, 2	mpancom 688	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑)
4		rexv 3488	. 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
5	3, 4	sylib 218	1 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4  ∃wex 1779   ∈ wcel 2108  ∃wrex 3060  Vcvv 3459  [wsbc 3765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-v 3461  df-sbc 3766

This theorem is referenced by:  spesbcd  3858  opelopabsb  5505  sbccomieg  42816  frege124d  43785  sbiota1  44458