Theorem sbcexf 38056

Description: Move existential quantifier in and out of class substitution, with an explicit nonfree 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 1913	. . . 4 ⊢ Ⅎ𝑧𝜑
2	1	sb8ef 2356	. . 3 ⊢ (∃𝑦𝜑 ↔ ∃𝑧[𝑧 / 𝑦]𝜑)
3	2	sbcbii 3827	. 2 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ [𝐴 / 𝑥]∃𝑧[𝑧 / 𝑦]𝜑)
4		sbcex2 3831	. 2 ⊢ ([𝐴 / 𝑥]∃𝑧[𝑧 / 𝑦]𝜑 ↔ ∃𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
5		sbcexf.1	. . . 4 ⊢ Ⅎ𝑦𝐴
6		nfs1v 2155	. . . 4 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑
7	5, 6	nfsbcw 3792	. . 3 ⊢ Ⅎ𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑
8		nfv 1913	. . 3 ⊢ Ⅎ𝑧[𝐴 / 𝑥]𝜑
9		sbequ12r 2251	. . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑 ↔ 𝜑))
10	9	sbcbidv 3826	. . 3 ⊢ (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ [𝐴 / 𝑥]𝜑))
11	7, 8, 10	cbvexv1 2342	. 2 ⊢ (∃𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
12	3, 4, 11	3bitri 297	1 ⊢ ([𝐴 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)

Syntax hints:   ↔ wb 206  ∃wex 1778  [wsb 2063  Ⅎwnfc 2882  [wsbc 3770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-v 3465  df-sbc 3771

This theorem is referenced by:  sbcexfi  38058