Theorem rspesbca 3832

Description: Existence form of rspsbca 3831. (Contributed by NM, 29-Feb-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
rspesbca	⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ 𝐵 𝜑)

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem rspesbca

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3745	. . 3 ⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
2	1	rspcev 3580	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑)
3		cbvrexsvw 3318	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 [𝑦 / 𝑥]𝜑)
4	2, 3	sylibr 237	1 ⊢ ((𝐴 ∈ 𝐵 ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ 𝐵 𝜑)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  [wsb 2101   ∈ wcel 2147  ∃wrex 3090  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-sbc 3743

This theorem is used by:  spesbc  3833  2nreu  4405  rexopabb  5509  indexfi  9387  indexdom  38936