Theorem sbcexf 34340

Description: Move existential quantifier in and out of class substitution, with an explicit non-free variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)

Hypothesis

Ref	Expression
sbcexf.1	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
sbcexf	⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem sbcexf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 2009	. . . 4 ⊢ Ⅎ𝑧𝜑
2	1	sb8e 2516	. . 3 ⊢ (∃𝑦𝜑 ↔ ∃𝑧[𝑧 / 𝑦]𝜑)
3	2	sbcbii 3652	. 2 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ [𝐴 / 𝑥]∃𝑧[𝑧 / 𝑦]𝜑)
4		sbcex2 3647	. 2 ⊢ ([𝐴 / 𝑥]∃𝑧[𝑧 / 𝑦]𝜑 ↔ ∃𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
5		sbcexf.1	. . . 4 ⊢ Ⅎ𝑦𝐴
6		nfs1v 2287	. . . 4 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
7	5, 6	nfsbc 3618	. . 3 ⊢ Ⅎ𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑
8		nfv 2009	. . 3 ⊢ Ⅎ𝑧[𝐴 / 𝑥]𝜑
9		sbequ12r 2279	. . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑 ↔ 𝜑))
10	9	sbcbidv 3651	. . 3 ⊢ (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]𝜑))
11	7, 8, 10	cbvex 2377	. 2 ⊢ (∃𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
12	3, 4, 11	3bitri 288	1 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 197  ∃wex 1874  [wsb 2062  Ⅎwnfc 2894  [wsbc 3596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-sbc 3597

This theorem is referenced by:  sbcexfi  34342