Theorem spesbc 3872

Description: Existence form of spsbc 3787. (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 3784	. . 3 ⊢ ([𝐴 / 𝑥]𝜑 → 𝐴 ∈ V)
2		rspesbca 3871	. . 3 ⊢ ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑)
3	1, 2	mpancom 687	. 2 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑)
4		rexv 3495	. 2 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
5	3, 4	sylib 217	1 ⊢ ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)

Syntax hints:   → wi 4  ∃wex 1774   ∈ wcel 2099  ∃wrex 3065  Vcvv 3469  [wsbc 3774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-v 3471  df-sbc 3775

This theorem is referenced by:  spesbcd  3873  opelopabsb  5526  sbccomieg  42135  frege124d  43114  sbiota1  43794