Theorem spesbc 3829

Description: Existence form of spsbc 3750. (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 3747	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
2		rspesbca 3828	. . 3 ⊢ ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑)
3	1, 2	mpancom 688	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑)
4		rexv 3465	. 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
5	3, 4	sylib 218	1 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4  ∃wex 1780   ∈ wcel 2113  ∃wrex 3057  Vcvv 3437  [wsbc 3737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rex 3058  df-v 3439  df-sbc 3738

This theorem is referenced by:  spesbcd  3830  opelopabsb  5475  sbccomieg  42913  frege124d  43881  sbiota1  44554