Theorem spesbc 3845

Description: Existence form of spsbc 3766. (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 3763	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
2		rspesbca 3844	. . 3 ⊢ ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑)
3	1, 2	mpancom 688	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑)
4		rexv 3475	. 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
5	3, 4	sylib 218	1 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053  Vcvv 3447  [wsbc 3753

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-v 3449  df-sbc 3754

This theorem is referenced by:  spesbcd  3846  opelopabsb  5490  sbccomieg  42781  frege124d  43750  sbiota1  44423