spesbc - New Foundations Explorer

	New Foundations Explorer		< Previous Next > Nearby theorems
Mirrors > Home > NFE Home > Th. List > spesbc		GIF version

Theorem spesbc 3127

Description: Existence form of spsbc 3058. (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 3055	. . 3 ⊢ ([̣A / x]̣φ → A ∈ V)
2		rspesbca 3126	. . 3 ⊢ ((A ∈ V ∧ [̣A / x]̣φ) → ∃x ∈ V φ)
3	1, 2	mpancom 650	. 2 ⊢ ([̣A / x]̣φ → ∃x ∈ V φ)
4		rexv 2873	. 2 ⊢ (∃x ∈ V φ ↔ ∃xφ)
5	3, 4	sylib 188	1 ⊢ ([̣A / x]̣φ → ∃xφ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∃wex 1541 ∈ wcel 1710 ∃wrex 2615 Vcvv 2859 [̣wsbc 3046

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047

This theorem is referenced by: spesbcd 3128 opelopabsb 4697